Cengage Maths Solutions Class 12 Inverse Trigonometric Functions

1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER INVERSE TRIGONOMETRIC FUNCTIONS || TRIGONOMETRY  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC Find the principal value of the following (ii) cosec− 1(2) tan−1( − √3) 1 1 cos−1( ) √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 2 Solve sin−1x ≻ 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 3 Solve cos−1x > cos−1x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 2− 3(cot−1x) + 2 > 0 4 Solve for if x (cot−1x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 5 Find the value of for which is defined. cosec−1(cosx) x  Watch Free Video Solution on Doubtnut

2. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC tan−1(x2) Find the value of (i) (ii) (iii) (iv) sin−1(2x) cos−1√x2− x + 1 1 + x2 6 1 sec−1(x + ) x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC Find the range of f(x) = ∣∣3tan−1x − cos−1(0)∣∣− cos−1( − 1) 7 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC π Find the value of for which sec−1x + sin−1x = 8 x . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC If sin−1(x2− 4x + 5) + cos−1(y2− 2y + 2) 9 =π then find the value of 2 xandy.  Watch Free Video Solution on Doubtnut

3. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 10 If then find the value of cos−1λ + cos−1μ + cos−1γ = 3π, λμ + μγ + γλ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC If sin−1x1+ sin−1x2+ + sin−1xn≤ − nπ , 2 11 n ∈ N, n = 2m + 1,m ≥ 1, x1 1+ x3 3+ x5 5+ ...(m + 1)terms then find the value of x2 4+ x6 2+ x4 6+ .... mterms  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 12 Find the values of for which will have at least one solution. sin−1x = |x − a| a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC Find satisfying integer function. where represents the greatest [] [tan−1x] + [cot−1x] = 2, x 13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC −1 −1

4. 14 Find the number of solution of the equation cos(cos−1x) = cosec(cosec−1x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC sinπ tanπ π 15 Evaluate the following: (ii) sin−1( cos−1(cos2 tan−1( ) ) ) 4 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC 2π 7π Evaluate the following: 1. 2. 3. sin− 1(sin cos−1(cos ) ) 3 6 16 2π tan− 1(tan ) 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC Find the value of sin−1(sin5) + cos−1(cos10) 17 + tan−1{tan( − 6)} + cot−1{cot( − 10)}.  Watch Free Video Solution on Doubtnut

5. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC Find the area bounded by and y = sin−1(sinx) 18 x = aξ or x ∈ [0,100π]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC 19 Find the number of solution of 2tan−1(tanx) = 6 − x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC 1 20 If then find the values of cos(2sin−1x) = , x. 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC 1 3 21 Find the value of sin( cot−1( − 2 )) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC 1 −1 22 Find the value of sin( cos−1( 4 )) 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC Prove that: 23

6. √1 + sinx + √1 − sinx x cot−1( ) = , x 2 √1 + sinx − √1 − sinx π ∈ (0, ) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC sin−1(1 − x) − 2sin−1x =π Solve 24 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC tan−1x Find in terms of where sin−1 25 x ∈ (0,a). √a2− x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 26 Simplify sincot−1tancos−1x, x > 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Prove that: cosec(tan−1(cos(cot−1(sec(sin−1a))))) 27 = √3 − a2, ∈ [0,1]

7. where a ∈ [0,1]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If then prove that cos−1x = π − sin−1√1 − x2 28 x < 0,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC cos−1x 1 + x 29 Prove that cos−1{ } = 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Prove that sin−1x 1 x tan− 1{ } = , − a 30 2 a + √a2− x2 a < x < a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Prove that: 31

8. √1 + x + √1 − x π sin−1{ } = 2 4 sin−1x + , 0 < x < 1 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 1 − x2n Prove that cos−1( ) = 2tan−1xn,0 < x < ∞ 32 1 + x2n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If then find the value of cos−1(2x2− 1) − 2sin−1x 33 x ∈ [ − 1,0],  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC π2 Find the minimum value of the function − cot−1x 34 f(x) = 16cot−1( − x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Find the range of y = (cot−1x)(cot−1( − x)). 35  Watch Free Video Solution on Doubtnut

9. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Find the value of sin−1(sin5) + cos−1(cos10) 36 + tan−1{tan( − 6)} + cot−1{cot( − 10)}.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC π If then find the value of sin−1x = cos−1x. 37 , f or somex ∈ ( − 1,1), 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 1 38 If then find the value of sin(sin−1( ) + cos−1x) = 1, x. 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 39 Solve sin−1x ≤ cos−1x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 40 Find the range of f(x) = sin−1x + tan−1x + cos−1x  Watch Free Video Solution on Doubtnut

10. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 2+ (c o s e c−1x) 2 41 Find the minimum value of (sec−1x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC sin−1(2√15) sin−1(14) 42 π Solve + = 2 |x| |x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC If α = sin−1(cos(sin−1x))andβ 43 = cos−1(sin(cos−1x)), . then find tanα tanβ  Watch Free Video Solution on Doubtnut

11. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC cos− 11 cos− 11 44 If then find the value of sec−1x = cosec−1y, + . y x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC Prove that: tan−11 π tan− 1x + = { , if x > 0 45 2 x π − , if x < 0 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC sin−11 cos−11 46 Find the value of sin−1x + + cos−1x + . x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 10 10 r Find the value of tan− 1( ). ∑ r= 1 ∑ s=1 47 s  Watch Free Video Solution on Doubtnut

12. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC If then find the maximum value of sin−1xi∈ [0, 1]∀i = 1,2,3,.28 √sin−1x1√cos−1x2 + √sin−1x2√cos− 1x3+ 48 √sin−1x3√cos−1x4 + + √sin−1x28√cos−1x1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC cos−1(12) cos− 1(33) cos−14 49 Prove that + = 5 13 65  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 50 If two angles of a triangle are then find the third angle. tan−1(2)andtan−1(3),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 51 Find the value of `tan^(-1)(1/2tan2A)+tan^(-1)(cotA)+tan^(-1)(cot^3A),for0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 52 Simplify `[(3sin2alpha)/(5+3cos2alpha)]+tan^(-1)[(tanalpha)/4],w h e r e-pi/2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC tan−12x + tan−13x =π Solve the equation 53 4  Watch Free Video Solution on Doubtnut

13. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 54 Solve tan−1x + sin− 1x = tan−12x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If cot− 1(xy + 1) then find the value of cot−1(yz + 1) x > y > z > 0, + x − y zy − z 55 cot− 1(zx + 1) + z − x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If is an A.P. with common difference then prove that a1,a2,a3,,an d, d tan[tan−1( ) 1 + a1a2 d + tan−1( ) 56 1 + a2a3 (n − 1)d d + tan−1( )] = 1 + an−1an 1 + a1an  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC ∞

14. 57 ∞ 1 Find the value of tan−1( ) ∑ 1 + r + r2 r= 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC tan−11 tan−11 tan−11 58 Find the value of 4 − + 5 70 99  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC tan−1(2x) 1 sin( − tan−1x) If , then find the value of (x − 1)(x2+ 1) > 0 59 1 − x2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC sin−15 cot− 13 60 Find the value of + 4 13  Watch Free Video Solution on Doubtnut

15. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC π Solve sin−1x + sin−12x = 61 . 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 62 Solve sin−1x ≤ cos−1x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Solve : x2 1 cos−1( ) x2+√1 − x2√1 − 2 4 63 cos−1x − cos−1x. = 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC π If then show that x ∈ (0, ), 2 7 cos−1( (1 + cos2x) + √(sin2x − 48cos2x)sinx) 2 64 = x − cos−1(7cosx)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Which of the following angles is greater? 4 1 θ1= sin−1( ) + sin−1( ) or θ2 5 3 65 4 1 = cos−1( ) + cos−1( ) 5 3  Watch Free Video Solution on Doubtnut

16. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Find the value lim n→ ∞ 66 n 1 +√(k − 1)k(k + 1)(k + 2) cos−1( ) ∑ k(k + 1) k=2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC If then prove that f(x) = sin−1x f(3x − 4x3) = π − 3 lim sin−1x lim 67 1 1 x→ x→ 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC Solve sin−1x − cos− 1x = sin− 1(3x − 2) 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC If 1 A = 2tan−1(2√2 − 1)andB = 3sin− 1( ) 3 69 3 + sin−1( ), 5 then which is greater.

17.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Some More Multiple Angle Formulas TRIGONOMETRIC sin−1(2x) tan−1(2x) 70 If , then find the value of = x. 1 + x2 1 − x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Some More Multiple Angle Formulas TRIGONOMETRIC 4x x 71 If is independent of ) find the values of sin−1( ) + 2tan−1( − x, x. x2+ 4 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Some More Multiple Angle Formulas TRIGONOMETRIC cos−1(6x) π 72 If then find the value of + tan−13x, = − x. 1 + 9x2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Some More Multiple Angle Formulas TRIGONOMETRIC cot−1(16) cos−13 cos−17 1 73 Find the value of 2 + + 63 2 25 √13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC

18. 74 Solve 2cos− 1x = sin−1(2x√1 − x2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC 1 + x2 75 Find the domain for f(x) = sin−1( ) 2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Find the sum cosec−1√10 + cosec−1√50 + cosec−1√170 76 + + cosec−1√(n2+ 1)(n2+ 2n + 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC Find the number cos−1y of positive sin−13 integral solution of the equation 77 tan− 1x + = √1 + y2 √10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC

19. 78 If `tan^(-1)y=4tan^(-1)x(|x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If x4− x3sin2β + x2cos2β − xcosβ − sinβ are the roots of the equations x1,x2, x3,andx4 = 0, prove that tan− 1x1+ tan−1x2 + tan−1x3 + tan−1x4 79 π = nπ + ( ) − β 2 , where is an integer. n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC 3+ (cos−1x)3 (sin−1x) 80 Solve for real values of x: = 7 (tan−1x + cot−1x)3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Complementary Angles TRIGONOMETRIC Find (sin−1x) the set of values 3= aπ3 of has a solution. parameter so that the equation a 81 3+ (cos− 1x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If and , then find the value of q − r p > q > 0 pr < − 1 < qr p − q tan− 1( ) + tan−1( ) 1 + qr 1 + qr r − p 82 + tan−1( ) 1 + qr  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC

20. Solve the equation √∣∣sin−1|cosx|∣∣+∣∣cos−1|sinx|∣∣ 83 −π = sin−1|cosx| − cos−1|sinx|, ≤ x 2 π ≤ . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC Solve the equation x + 1 x − 1 tan− 1( ) + tan−1( ) 84 x − 1 x = tan−1( − 7)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC −π 85 Solve the equation sin−16x + sin−16√3x = . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If then prove that 0 < a1< a2< .... < an,

21. a1x − y a2− a1 86 tan− 1( ) + tan−1( ) x + a1y 1 + a2a1 a3− a2 + tan−1( ) + ....... 1 + a3a2 an− an−1 1 + tan−1( ) + tan−1( ) 1 + anan−1 an x = tan−1( ). y  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC Let f(x) = sinx + cosx + tanx + sin−1x 87 + cos−1x + tan−1x. Then find the maximum and minimum values of f(x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 3π π is equal to (b) (c) (d) cos−1(cos(2cot−1(√2 − 1))) √2 − 1 88 4 4 noneofthese  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of sin−1 ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ sin−1⎛ cos−1(√12) 2 − √3 ⎜ ⎝ cot + 4 4 89 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ π ⎞ ⎟ ⎠ + sec−1√2 is π (a) (b) 0 (c) (d) none of these 2 3  Watch Free Video Solution on Doubtnut

22. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 2 √6 + 1 π π π π − cos−1( is equal to ) The value of (b) (c) (d) cos−1√ 90 3 3 4 2 6 2√3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 1 1 3 −3 1 1 91 The value of is (a) (b) (c) (d) cos( cos−1( 2 )) 8 4 4 16 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC cot−1n π If then the maximum value of is 6 (b) (c) 5 (d) none of 92 > ,n ∈ N, 7 n 6 π these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If and are equal functions, then the maximum cosec−1(cosecx) cosec(cosec−1x) π π π π 93 range of value of is (a) x (b) (c) [ − , − 1] ∪ [1, ] [ − , 0] ∪ [0, ] 2 2 2 2 (d) ( − ∞, − 1) ∪ [1,∞] [ − 1, 0] ∪ [0,1]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 2 01 2 −1

23. 94 is equal to 5 (b) 13 (c) 15 (d) 6 sec2(tan012) + cosec2(cot−13)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Some More Multiple Angle Formulas TRIGONOMETRIC f(x) = tan−1⎛ ⎞ ⎟ ⎠ (√12 − 2)x2 The maximum value of is ⎜ ⎝ 95 x2+ 2x2+ 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC For the equation (b) 2 (c) 0 (d) , the number of real solution is (a)1 cos−1x + cos−12x + π = 0 96 ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC The number of real solution of the equation tan− 1√x2− 3x + 7 + cos−1√4x2− x + 3 97 = π is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC If 98

24. sin−1(x − 1) + cos−1(x − 3) x + tan−1( ) = cos−1k + π, 2 − x2 1 1 then the value of is (a)1 (b) (c) (d) non of these − k √2 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If (b) then is equal to sin−1x = θ + βandsin−1y = θ − β, sin2θ + cos2β cos2θ + cos2θ sin2θ + sin2β 1 + xy 99 (d) cos2θ + sin2β  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC 3π 3π π π Range of is (a) (b) f(x) = sin−1x + tan−1x + sec−1x ( ) [ ] , , 4 4 4 4 100 3π π (c) (d) none of these } { , 4 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 1 1 π The value of is equal to (b) (c) (d) ∞(tan−1x))) 101 ( lim )n− → −1 − 2 √2 √2  Watch Free Video Solution on Doubtnut

25. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC 2x π π π π π π Range of is (a) (b) (c) (d) ) tan−1( ) [ − ] ( − ) ( − , , , 1 + x2 102 4 4 2 2 2 4 π π [ ] , 4 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC If then the complete set of values of is of these , where denotes the greatest integer functions, (b) x (cos1, 1) cos1,cos1) (cot1,1) [cot−1x] + [cos−1x] = 0 [] 103 (d) none  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC Complete solution set of greatest integer function, is equal to (a) where denotes the [] [cot−1x] + 2[tan−1x] = 0, 104 (b) (d) (0, cot1) (0, tan1) (tan1,∞) (cot1, tan1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC The sin−1x + tan−1x = 2k + 1 number of integral values has a solution is 1 (b) 2 (c) 3 (d) 4 of for which the equation k 105  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The range of value of for which the equation p has a sincos−1(cos(tan−1x)) = p 106 1 1 1 solution is (a) (b) (0,1) ) (d) ( − (c)( , 1) , ( − 1, 1) √2 √2 √2  Watch Free Video Solution on Doubtnut

26. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC The sum of the solution of the equation 2sin−1√x2+ x + 1 + cos−1√x2+ x 107 =3π 2−1 is 0 (b) (c) (d) 2 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 1 1 Complete solution set of is (a) tan2(sin−1x) > 1 ( − 1, − ) ∪ ( ,1) √2 √2 108 1 1 (b) (c) (d) none of these ( − )~{0} , ( − 1, 1)~{0} √2 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of 4π − 24 is equal to (a) (b) sin−1(sin12) + cos−1(cos12) 24 − 2π zero 109 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of the expression sin(22π) cos(5π) sin−1( ) + cos−1( ) 7 3 110 tan(5π) ) + sin−1(cos2) + tan−1( 7 17π −

27. 17π −π is (a) (b) (c) (d) − 2 −2 − 2 noneofthese 42 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC π The value of where , is sin−1(cos(cos− 1(cosx) + sin−1(sinx))), x ∈ ( , π) 111 2 π −π equal to (b) (c) (d) −π π 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 3π If , then the value of , − π) α ∈ ( − 2 112 tan− 1(cotα) − cot− 1(tanα) + sin−1(sinα) + cos−1(c0sα) is equal to (a) (b) (c) 0 (d) 2π + α π + α π − α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC cosx 3π π x π is equal to tan− 1[ ] , f or x ∈ ( − ) − , 1 + sinx 4 2 2 2 π x π π π x π π 113 ,f or x ∈ ( − ) ,f or x ∈ ( − ) − , − , 4 π 2 x 2 3π 2 4 2 2 2 π ,f or x ∈ ( − ) − , 4 2 2 2  Watch Free Video Solution on Doubtnut

28. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC If f(tan−1(tan8) and is constant, then f(x) = x11+ x9− x7+ x3+ 1 f(sin−1(sin8) = α, α 114 is equal to (b) (c) (d) α − 2 α + 2 2 − α α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC The trigonometric equation |a| <1 a|a| ≤ has a solution for all real values (b) sin−1x = 2sin−1a 115 1 1 1 (d) < |a| < 2 √2 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC hold if (a) (b) (c) sin−1(sin5) > x2− 4x x = 2 − √9 − 2π x = 2 + √9 − 2π 116 (d) x > 2 + √9 − 2π x ∈ (2 − √9 − 2π,2 + √9 − 2π)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of tan(sin− 1(cos(sin−1x)))tan 117 (cos−1(sin(cos−1x))), wherex ∈ (0, 1), is equal to 0 (b) 1 (c) (d) none of these −1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC There exists a positive real number of satisfying Then the cos(tan−1x) = x. x 2 2 4

29. 118 x2 2π 4π π π value of (b) (c) (d) cos−1( )is 2 10 5 5 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of α3 tan−1α 1 cosec2( ) 2 2 β 119 β3 1 β sec2( tan−1( 2 (α + β)(α2+ β2) ))isequa < o + 2 α (b) (d) none of these (α + β)(α2− β2) (α + β)(α2+ β2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 1 + x4+ y4 1 π If is equal to 1 (b) 2 (c) (d) sin−1x + sin−1y = , then 120 x2− x2y2+ y2 2 2 none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC The value of cot−11 is equal to (a) (b) 2tan− 1(cosectan−1x − tancot−1x) cot−1x 121 (c) (d) none of these tan−1x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC

30. tan−1(√1 + x2− 1) x = tan40x =tan1 If then (b) (d) = 40 x = tan20 40 122 x x = tan80  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If then the value of sin−1a + sin− 1b + sin−1c = π, 1 will be (b) (c) (d) a√(1 − a2) + b√(1 − b2) + √(1 − c2) 2abc abc abc 123 2 1 abc 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If πab + c(b − a) then is equal to (a) (b) asin−1x − bcos−1x = c, asin−1x + bcos−1x πab + c(a − b) 0 124 π (c) (d) a + b 2 a + b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC 0, 1 The solution of the inequality is (b) ] x ∈ [ (log)sin−1x> (log)1/2cos−1x 2 1 √2 125 1 0,1 (d) none of these ) x ∈ [ , 1] x ∈ ( √2 √2  Watch Free Video Solution on Doubtnut

31. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC For , is true when belongs to (a) θ sin−1(sinθ) > cos−1(sinθ) 0 < θ < 2π 3π 3π 3π π π 126 (b) (c) (d) ) (π, ) ( ( , 2π) ( , π) , 4 2 4 4 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC π If (a) (b) (c) (d) ∣∣sin−1x∣∣+∣∣cos−1x∣∣= 127 , then x ∈ R [ − 1,1] [0, 1] φ 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Mean and standard deviation of 100 observations of are found to be 40 and 10. If at the time of calculation two observations are wrongly taken as 30 and 70 instead of 3 and 27 respectively. Find the correct standard deviation. 128  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC sin− 1|x − 2| + cos−1(1 − |3 − x|) =π The number of integer satisfying is x 129 2 (a) (b) (c) (d) 1 2 3 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC

32. 130 2π √3 − 1 If then is equal to (a) (b) 3 (c) (d) tan−1x + 2cot−1x = , x, √3 3 √3 + 1 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC The number 1 + x2 of solutions of the equation π 131 is 0 (b) 1 (c) 2 (d) 3 cos−1( ) − cos−1x = + sin−1x 2 2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 1 f(x) = tan−1x + tan− 1( );g(x) x 132 = sin−1x + cos−1x are identical functions if (a) (b) (c) (d) x ∈ R x > 0 x ∈ [ − 1, 1] x ∈ [0,1]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC The value of for which ax2+ sin− 1(x2− 2x + 2) a 133 + cos−1(x2− 2x + 2) = 0 2 −2 π −π has a real solution is (b) (c) (d) 2 2 π π  Watch Free Video Solution on Doubtnut

33. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 5 12 7 4 13 π 134 If then is equal to x (b) (c) (d) sin−1( ) + sin−1( ) = , 13 2 13 3 7 x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If cos−1√p + cos−1√1 − p + cos−1√1 − q 3π 135 = , 4 1 1 1 then the value of is 1 (b) (c) (d) q 3 3 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC −π π 3+ (cos−1x) 3 The least and the greatest values of are (a) (b) (sin−1x) , 2 2 136 −π3 π3 π3 7π3 (c) (d) none of these , , 8 8 32 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC If tan− 1(sin2θ − 2sinθ + 3) 137 π + cot−1(5sec^ (2y) + 1) = , 2 then value of is equal to 0 (b) (c) (d) none of these −1 1 cos2θ − sinθ  Watch Free Video Solution on Doubtnut

34. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The product of all values of satisfying the equation 2x2+ 10|x| + 4 x sin−1cos( ) x2+ 5|x| + 3 138 2 − 18|x| π = cot(cot−1( )) + is 2 9|x| −1 9 (b) (c) (d) −9 −3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC x − y 3π x π π π π 139 is (b) (c) (d) tan− 1( ) − tan−1( ) or x + y 2 3 4 4 4 y  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC If tan−1(a + x) tan−1(a − x) π + = , 140 6 a a thenx2= 2√3a (b) (c) (d) none of these 2√3a2 √3a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC 141 If `y=tan^(-1)1/2+tan^(-1)b ,(0

35.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If number of ordered triplets solutions are natural numbers such that then the cot−1x + cot− 1y = cot−1z x, y,z 142 that satisfy the equation is 0 (b) 1 (c) 2 (d) Infinite (x,y,z)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC sin−12 sin−13 The value of such that are the angles of a triangle is , sin−1α , α √5 √10 143 −1 1 1 1 (b) (c) (d) 2 √2 √3 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC tan−1(1 + x) + tan−1(1 − x) =π The number of solutions of the equation is 2 144 2 (b) 3 (c) 1 (d) 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If cot−1x + cot−1y + cot−1z = π ,x,y,z 2 145 > 0andxy < 1, 1 1 1

36. 1 1 1 then is also equal to (b) (d) none of x + y + z + + xyz xy + yz + zx y x z these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If x2+ y2+ z2+ 2xyz = 0 x2+ y2+ z2+ xyz = 1 x2+ y2+ z2+ 2xyz = 1 cos−1x + cos−1y + cos−1z = π, then x2+ y2+ z2+ xyz = 0 146  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC π If tan−1x + tan− 1y + tan−1z = , then x + y + z − xyz = 0 147 2 x + y + z + xyz = 0 xy + yz + zx + 1 = 0 xy + yz + zx − 1 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If xy x2+ y2+ z2= r2, thentan−1( ) zr 148 yz xz + tan−1( ) + tan−1( ) xr yr π is equal to (b) (c) 0 (d) none of these π 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC n √r − √r − 1 tan−1(√n) −π sin−1( )isequa < o ∑ 149 4 √r(r + 1) r=1 tan− 1(√n + 1) −π (d) 4tan−1(√n) tan−1(√n + 1)  Watch Free Video Solution on Doubtnut

37. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC The value of tan−14 tan−14 tan−14 tan−14 + + + 7 19 39 67 + ∞equals 150 tan−11 tan−11 tan−1+ cot−13 cot−11 +cot−11 cot−11 tan− 11 + + 2 3 2 3 cot−11 + tan−13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC The sum of series sec− 1(√10) sec−1(√50) sec−1√2 + + 3 7 151 (n2+ 1)(n2− 2n + 2) + + sec−1√ (n2− n + 1)2 is (b) (d) tan− 11 n tan−1(n + 1) tan−1(n − 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 1 π cos−1x) tan( + 4 2 152 1 π + tan( cos−1x), x ≠ 0, − 4 2 2 is equal to (b) (c) (d) none of these 2x x x

38.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC xcosθ cosθ θ The value of (b) (c) θ (d) tan−1( ) − cot−1( )is 2θ 153 1 − xsinθ x − sinθ 2 independent of θ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC tan2α cot2α If then is (b) (c) cot−1(√cosα) − tan−1(√cosα) = x, sinx 2 2 154 cotα (d) tan2α 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC 1 3sin2θ If (a) (b) (c) sin−1[ ] = tan−1x,thenx = tan3θ 3tanθ 2 5 + 4cos2θ 155 1 (d) ( )tanθ 3cotθ 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC −

39. 156 a − b tanθ acosθ + b The value is equal to (b) ) 2tan−1[√ cos−1( ] a + b 2 a + bcosθ a + bcosθ acosθ bcosθ (d) ) cos−1( cos− 1( cos−1( ) ) a cosθ + b a + bcosθ a cosθ + b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC tan−11 tan−11 1 If then is equal to 1 (b) 2 (c) 3 (d) x 3tan−1( ) − = , 157 3 x 2 + √3 √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC If 2a 2b sin−1( ) + sin−1( ) 1 + a2 1 + b2 158 = 2tan−1x, a − b a + b b b then is equal to x (a) (b) (c) (d) [a, b, ∈ (0, 1)] 1 + ab 1 + ab 1 + ab 1 − ab  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If 1 − x2 2x 3sin−1( ) − 4cos−1( ) 1 + x2 1 + x2 159 2x π + 2tan−2( ) = ,where|x| < 1, 1 − x2 3 1 1 −√3 then is equal to x (b) (c) (d) − √3 4 √3 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If 1 + x x1= 2tan−1( ),x2 1 − x 160 1 − x2 = sin−1( ) 1 + x2 , where then is equal to 0 (b) (c) (d) none of these π x ∈ (0,1), x1+ x2 2π

40.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC If u = cot−1√tanα − tan−1√tanα, 161 π u thentan( )isequa < o , 4 2 (a) (b) (c) (d) cot √tanα √cotα tanα α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC 162 If the equation `x^3+b x^2+c x+1=0,(b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 163 The value of is equal to sin−1[x√1 − x −√x√1 − x2]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If sin−1x + sin−1y + sin−1z = π,thenx4 + y2+ z4+ 4x2y2z2= K(x2y2+ y2z2 164 + z2x2), where is equal to 1 (b) 2 (c) 4 (d) none of these K

41.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If √3 1 1 f(x) = sin−1( √1 − x2), − x − 165 2 2 2 ≤ x ≤ 1,thenf(x) is equal to  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC tan−1x tan−12x tan−13x ∣∣∣∣∣ ∣∣∣∣∣ Let =0 , then the number of values of satisfying tan−13x tan−1x tan−12x x 166 tan−12x tan−13x tan−1x the equation is 1 (b) 2 (c) 3 (d) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC Which of the following 2x − 1 is the solution set of the equation (a)(0.1) (b) (c) (d) 2cos−1x = cot−1( )? 167 ( − 1, 1) − {0} ( − 1,0) 2x√1 − x2 ( − 1,1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC

42. FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 The number of solution of equation sin−1x + nsin− 1(1 − x) = mπ 168 , wheren 2 > 0,m ≤ 0, is 3 (b) 1 (c) 2 (d) None of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC The solution set of the equation sin−1√1 − x2+ cos−1x cot−1(√1 − x2) 169 − sin−1x = x is (a) (b) (c) (d) [ − 1,1] − {0} (0,1) ∪ { − 1} ( − 1,0) ∪ {1} [ − 1, 1]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC π The equation has one negative solution one positive solution 3−1x − πx − = 0 170 2 no solution more than one solution  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Angle S In Terms Of Sin^-1X And Cos^-1X TRIGONOMETRIC 171 If `|cos^(-1)((1-x^2)/(1+x^2))|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC If 22π/sin( −1)x− 2(a + 2)π/sin( −1)x+ 8a < 0 172 1 for at least one real then (b) (d) x, ≤ a < 2 a < 2 a ∈ R − {2} 8 1 a ∈ [0, ] ∪ (2, ∞) 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC 2

43. 173 If following real? (a) are the roots of equation (b) cos−1α , then which and cot−1β 6x2+ 11 = x + 3 = 0 cosec−1α α, β(α < β) (c) (d) both sin−1β cot−1α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Multiple Anlges In Terms Of Tan^-1X TRIGONOMETRIC −π +cos−13 −3 is equal to (a) (b) ) (c) 2tan−1( − 2) −cost−1( 5 5 174 3 3 π (d) ) + tan−1( − −πcot−1( − ) − 2 4 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 175 If is equal to x < 0,thentan−1x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC 1 If , then is equal to (a) (b) ) cos−1x sec−1( π − sin−1√1 + x2 −1 < x < 0 x 176

44. x x (c) (d) . π + tan−1( cot−1( ) ) √1 − x2 √1 − x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If (sin−1x + sin−1w)(sin−1y + sin−1z) 177 = π2, then maximum value of 0 16 different D are possible has a minimum value of has a maximum value of 2 has a D = ∣∣xN1yN3zN3wN4∣∣(N1,N2,N3, N4∈ N) −2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Indicate the relation which can hold in their respective domain for infinite values of (b) tan∣∣tan−1x∣∣= |x| cot∣∣cot−1x∣∣= |x| sin∣∣sin−1x∣∣= |x| x. (d) tan−1|tanx| = |x| 178  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Inverse Trigonometirc Functions TRIGONOMETRIC n2− 10n + 21.6 π If where then the possible values of xy < 0 cot−1( ) > , z 179 6 π is (are) 3 (b) 2 (c) 4 (d) 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC

45. 1 1 If where then the possible z = sec−1(x + ) + sec−1(y + ), xy < 0, 180 y x 8π 7π 9π 21π values of is (are) (b) (c) (d) z 10 10 10 20  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC The value of such that the length of the longest interval in which the k(k > 0) π 181 function is constant is is/ are (a)8 (b) 4 f(x) = sin−1|sinkx| + cos− 1(coskx) 4 (c) 12 (d) 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Principal Value Of The Function TRIGONOMETRIC Which y = tan(cos−1x); y =√1 − x2 of the following pairs of function/functions has same graph? y = tan(cot− 1x);y =1 182 x x x y = sin(tan−1x); y = √1 − x2y = cos(tan−1x);y = s ∈ (cot−1x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC π2 1 sin−1x + sin−1y =π π If and then (a) + √ sin2x = cos2y, x = − 64 2 8 2 183 π2 π2 π2 1 1 1 π π π (b) (c) (d) y = √ + √ y = √ − − x = − − − 64 64 64 2 12 12 2 2 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If 2(sin−1x + sin−1y + sin−1z) = cos−1x cos−1x + cos−1y + cos−1z = π, then x2+ y2+ z2+ 2xyz = 1 184 + cos−1y + cos−1z 1 1 1 xy + yz + zx = x + y + z − 1 (x + ) + (y + ) + (z + ) ≥ 6 y x z  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC

46. FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` If tan− 1(x2+ 3|x| − 4) 185 π + cot−1(4π + sin−1sin14) = ,then 2 the value of is (a) (b) (c) (d) sin−1sin2x 6 − 2π 2π − 6 π − 3 3 − π  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC sin−1(2x) 186 If is independent of then (b) (c) `0 2tan−1x + x, x > 1 x < − 1 1 + x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC If 4x − 4x3 α = tan−1( ), β 1 − 6x2+ x2 2x = 2sin−1( ) 187 1 + x2 tanπ 1, 1 and then (a) for (b) for x ∈ [ ] = k, α + β = π α + β x ∈ ( − k,k) 8 k 1, 1 (c) for (d) for x ∈ [ ] α + β = π α + β = 0 x ∈ [ − k,k] k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC 2tan(tan−1(x) + tan− 1(x3)), wherex ∈ R 188 − { − 1,1}, 2x

47. 2x is equal to 1 − x2t(2tan− 1x) tan(cot−1( − x) − cot−1(x)) tan(2cot−1x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC Let 4 36 α = som−1( ),β = cos− 1( )andγ 85 5 8 = tan−1( ) 189 15 then tanαtanβ + tanβtanγ + tanαtanγ = 1 tanα + tanβ + tanγ = tanα tanβtanγ cotα cotβ + cotβcotγ + cotαcotγ = 1 cotα + cotβ + cotγ = cotα cotβcotγ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If Sn= cot−1(3) + cot−1(7) + cot−1(13) ..n + cot−1(21) + 190 5 4 S∞=π terms, then (c) (d) S10= tan−1( S6= sin−1( ) ) 6 4 5 S20= cot−11. 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC Equation (b) exactly two value of (c) exactly one value of (d) exactly two value of x is satisfied by (a) exactly one value of 1 + x2+ 2xsin(cos−1y) = 0 x 191 y y  Watch Free Video Solution on Doubtnut

48. CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC 1 )2π/cos−1x− a2= 0 To the equation has only one real 22π/cos−1x − (a + 192 2 root, then (a) (b) (c) (d) 1 ≤ a ≤ 3 a ≥ 1 a ≤ − 3 a ≥ 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC The solution set of inequality π (cot−1x)(tan−1x) + (2 − )cot− 1x 2 193 π − 3tan−1x − 3(2 − ) > 0 cot−1a + cot−1b 2 is then the value of is____ (a, b),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC If x = sin−1(a6+ 1) + cos−1(a4+ 1) 194 − tan−1(a2+ 1),a ∈ R, then the value of is_______ sec2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC The number of values of for which x 195 4 6

49. x4 x6 sin−1(x2− ) + 3 9 x8 x12 π + cos−1(x4− ..) = + , 3 9 2 where `0lt=|x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC 196 If `0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Tan^-1` TRIGONOMETRIC If 3 3 tan− 1(x + ) − tan−1(x − ) x x tan−16 197 = , x then the value of is_____. x4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC If range of function the value of is then f(x) = sin− 1x + 2tan−1x + x2+ 4x + 1 [p,q], 198 is_______> (p + q)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC

50. FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 If cos−1(x) + cos−1(y) + cos−1(z) 199 = π(sec2(u) + sec4(v) + sec6(w)), whereu, v, w are least non-negative angles such that `u  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Absolute value of sum of all integers in the domain of f(x) = cot−1√(x + 3)x 200 + cos−1√x2+ 3x + 1 is_______  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Miscellaneous TRIGONOMETRIC The least value of is ________ (1 + sec−1x)(1 + cas−1x) 201  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Sum And Difference Of Angles In Term Of Sin^-1 And Cos^-1 TRIGONOMETRIC Let ax3+ bx2+ cx − c1= 0, satisfies the equation is_________ (b − a − c) If cos−1(x) + cos−1(2x) + cos−1(3x)beπ. x 202 then the value of  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_INVERSE FUNCTIONS_Relating Different Inverse Trigonometric Functions TRIGONOMETRIC Number of solutions of equation sin(cos− 1(tan(sec−1x))) = √1 + xis/are_ 203 _  Watch Free Video Solution on Doubtnut