# Inverse Trigonometric Functions Jee Main and Advanced Questions

## Multiple Choice Questions

Question 1
if $sin^{-1} x + sin^{-1} y + sin^{-1} z = \frac {3 \pi}{2}$, then the value of $x^{21} +y^{21} + z^{21} - 3xyz$ is
(a) 0
(b) 1
(c) -1
(d) None of these

Question 2
if $x \geq 1$, then the value $2tan^{-1}x + sin^{-1} \frac {2x}{1+x^2}$ is ?
(a) $\pi$
(b) $2 \pi$
(c) $0$
(d) $-\pi$

Question 3
Two angles of triangle are $tan^{-1} 1/2$ and $cot^{-1} 3$, then the third angle is ?
(a) $\pi/2$
(b) $3\pi/4$
(c) $\pi/6$
(d) $\pi/3$

Question 4
The equation $2sin^{-1} x + cos^{-1} =\frac {11 \pi}{6}$ has ?
(a) 1 solution
(b) 2 solution
(c) Infinite solutions
(d) No solutions

Question 5
The value of $cot (\sum_{n=1}^{23}cot^{-1}\left ( 1+\sum_{k=1}^{n} 2k\right ))$ ?
(a) 23/25
(b) 25/23
(c) 24/23
(d) 23/24

Question 6
The number of real solutions of $tan^{-1} \sqrt {x(x+1)} + sin^{-1} \sqrt {x^2 +x + 1} = \frac {\pi}{2}$ is
(a)1
(b)0
(c)2
(d) 3

Question 7
If $sin^{-1} \frac {x}{5} + cosec^{-1} \frac {5}{4} = \frac {\pi}{2}$ , then value of x is
(a) 1
(b) 3
(d) 4
(d) 2

Question 8

The value of $sin^{-1} sin12 + cos^{-1} cos 12$ then
(a) 0
(b) 12
(c) 24
(d) -12

Question 9
The greatest and least value of the function $f(x) = (sin^{-1}x)^2 + (cos^{-1}x)^2$ are
(a)$\frac{\pi ^2}{4}$, 0
(b) $\frac{5\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(c) $\frac{\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(d) $\frac{3\pi ^2}{4}$, $\frac{\pi ^2}{8}$

Question 10
The value of $sin^{-1} ( cos \frac {33\pi}{5})$ is
(a) $\frac {3\pi}{5}$
(b) $-\frac {\pi}{10}$
(c) $\frac {\pi}{10}$
(d) $\frac {7\pi}{5}$

1. a
as $sin^{-1} x \leq \frac {\pi}{2}$
so, $sin^{-1} x=\frac {\pi}{2}$, $sin^{-1} y=\frac {\pi}{2}$,$sin^{-1} z=\frac {\pi}{2}$
or x=y=z=1
$x^{21} +y^{21} + z^{21} - 3xyz=0$
2. a
3. b
4. d
5. b
$cot (\sum_{n=1}^{23}cot^{-1}\left ( 1+\sum_{k=1}^{n} 2k\right ))= cot(\sum_{n=1}^{23}cot^{-1} ( 1+ 2\frac {n(n+1)}{2} ))$
$=cot(\sum_{n=1}^{23}cot^{-1} [ 1+ n(n+1)])= cot(\sum_{n=1}^{23}tan^{-1} \frac {1}{ 1+ n(n+1)})$
$=cot(\sum_{n=1}^{23} [ tan^{-1} (n+1) - tan^{-1} n]) =cot(tan^{-1} 24 - tan^{-1} 1) = cot tan^{-1} \frac {23}{25} =\frac {25}{23}$
6. c
7. b
$sin^{-1} \frac {x}{5} + cosec^{-1} \frac {5}{4} = \frac {\pi}{2}$
$sin^{-1} \frac {x}{5} + sin^{-1} \frac {4}{5} = \frac {\pi}{2}$
$sin^{-1} \frac {x}{5}= \frac {\pi}{2} - sin^{-1} \frac {4}{5}$
$sin^{-1} \frac {x}{5}=cos^{-1} \frac {4}{5}$
$sin^{-1} \frac {x}{5}=sin^{-1} \frac {3}{5}$
x=3
8. a
$sin^{-1} sin12 + cos^{-1} cos 12= 12 - 3 \pi + 3 \pi -12=0$
9. b
$f(x) = (sin^{-1}x)^2 + (cos^{-1}x)^2 = (sin^{-1} x + cos^{-1} x) ^2 - 2 sin^{-1} x cos^{-1} x$

$= \frac {\pi^2}{4} - 2 sin^{-1} x ( \frac {\pi}{2} - sin^{-1} x)= \frac {\pi^2}{4} - \pi sin^{-1} x +2 (sin^{-1} x)^2$
$= 2((sin^{-1} x)^2 - \frac {\pi }{2} sin^{-1} x \frac {\pi^2}{8})= 2[(sin^{-1} x - \frac {\pi}{4})^2 + \frac {\pi^2}{16}]$
So greatest and least value of the function are $\frac{5\pi ^2}{4}$, $\frac{\pi ^2}{8}$
10. b
$sin^{-1} ( cos \frac {33\pi}{5})= sin^{-1} [ cos (6\pi + \frac {3\pi}{5})]= sin^{-1} cos( \frac {3\pi}{5}) = sin^{-1} sin(\frac {\pi}{2}- \frac {3\pi}{5})$
$=sin^{-1} sin(\frac {-\pi}{10})=\frac {-\pi}{10}$

## Numerical values questions

Question 11
(i) Considering only the principal valeus of the inverse trigonometric function , the value of the expression
$\frac {3}{2} cos^{-1} \sqrt {\frac {2}{2 + \pi ^2}} +\frac {1}{4}sin^{-1} \frac {2\sqrt 2 \pi}{2 + \pi^2} + tan^{-1} \frac {\sqrt 2}{\pi}$ is ________
(ii) The principal value of $tan^{-1} \sqrt 3$ is _____
(iii) The number of real solution of the equation
$sin^{-1}\left ( \sum_{i=1}^{\infty }x^{i+1} - x \sum_{i=1}^{\infty } (x/2)^i\right )=\frac {\pi}{2} - cos^{-1}\left ( \sum_{i=1}^{\infty }(-x/2)^{i} - \sum_{i=1}^{\infty } (-x)^i\right )$ lying in the interval (-1/2,1/2) is ________
(iv) The value of $sec^{-1} \left ( \frac {1}{4} \sum_{k=0}^{10} sec \left ( \frac {7\pi}{12} + \frac {k \pi}{2} \right ) sec \left ( \frac {7\pi}{12} + \frac {(k+1)\pi}{2} \right ) \right )$ in the interval $[-\frac {\pi}{4}, \frac {3 \pi}{4}]$ is _____

(i) $\frac {3\pi}{4}$
let $tan \theta =\frac {\pi}{\sqrt 2}$ or $\pi \sqrt 2 tan \theta$
So $\frac {\pi}{4} < x < \frac {\pi}{2}$
so,$\frac {3}{2} cos^{-1} \sqrt {\frac {2}{2 + \pi ^2}} +\frac {1}{4}sin^{-1} \frac {2\sqrt 2 \pi}{2 + \pi^2} + tan^{-1} \frac {\sqrt 2}{\pi}$
$=\frac {3}{2} cos^{-1} \sqrt {\frac {2}{2 + 4tan^2 \theta}} +\frac {1}{4}sin^{-1} \frac {2\sqrt 2 (2 tan \theta)}{2 + 4tan^2 \theta} + tan^{-1} cot \theta$
$=\frac {3}{2} cos^{-1} cos \theta + \frac {1}{4} sin^{-1} sin 2 \theta + tan^{-1} cot \theta$
$=\frac {3}{2} \theta + \frac {1}{4} (\pi -2 \theta) + \frac {\pi}{2} - \theta= \frac {3\pi}{4}$

(ii)$\frac {\pi}{3}$
(iii) 2
$sin^{-1}\left ( \sum_{i=1}^{\infty }x^{i+1} - x \sum_{i=1}^{\infty } (x/2)^i\right )=\frac {\pi}{2} - cos^{-1}\left ( \sum_{i=1}^{\infty }(-x/2)^{i} - \sum_{i=1}^{\infty } (-x)^i\right )$
$sin^{-1}\left ( \frac {x^2}{1-x} - \frac {x(x/2)}{1-x/2}\right )= \frac {\pi}{2} - cos^{-1}\left ( \frac {-x/2}{1+x/2} - \frac {-x}{1+x}\right )$
$sin^{-1}\left ( \frac {x^2}{(1-x)(2-x)} \right ) =\frac {\pi}{2} - cos^{-1} \left ( \frac {x}{(1+x)(2+x)}\right )$
$sin^{-1}\left ( \frac {x^2}{(1-x)(2-x)} \right ) =sin^{-1} \left ( \frac {x}{(1+x)(2+x)}\right )$
$\frac {x^2}{(1-x)(2-x)}= \frac {x}{(1+x)(2+x)}$
$x(x^3 + 2x^2 + 5x -2)=0$
so x =0 or $x^3 + 2x^2 + 5x -2=0$
Now $x^3 + 2x^2 + 5x -2=0$ is increasing function for all x
Also f(0)=-2 , f(1/2) > 0
So there lies one value in (0, 1/2) for which it will become zero
Hence total number of solutions =2
(iv) 0
$sec^{-1} \left ( \frac {1}{4} \sum_{k=0}^{10} sec \left ( \frac {7\pi}{12} + \frac {k \pi}{2} \right ) sec \left ( \frac {7\pi}{12} + \frac {(k+1)\pi}{2} \right ) \right )$
$=sec^{-1} \left ( -\frac {1}{4}\sum_{k=0}^{10} sec \left ( \frac {7\pi}{12} + \frac {k \pi}{2} \right ) cosec \left ( \frac {7\pi}{12} + \frac {k\pi}{2} \right ) \right )$
$=sec^{-1} \left ( -\frac {1}{4} \sum_{k=0}^{10} \frac {1}{sin \frac {7\pi}{6}. (-1)^k} \right )$
$=sec^{-1} \left (-\frac {1}{2} \frac {1}{sin \frac {7\pi}{6}} \right )$
$=sec^{-1} 1 =0$

Question 12

p -> b
q -> d
r -> a
s -> c

## Subjective Questions

Question 13
Prove that
$tan^{-1} \sqrt {\frac {a(a+b+c)}{bc}} + tan^{-1} \sqrt {\frac {b(a+b+c)}{ac}} + tan^{-1} \sqrt {\frac {c(a+b+c)}{ab}} = \pi$

Question 14
if $cos^{-1} \frac {x}{a} + cos^{-1} \frac {y}{b} =\alpha$
Prove that
$\frac {x^2}{a^2} - \frac {2xy}{ab} cos \alpha + \frac {y^2}{b^2} = sin^2 \alpha$

Question 15
Solve the equation
$tan^{-1} (x + \frac {2}{x}) - tan^{-1} (x - \frac {2}{x}) - tan^{-1} \frac {4}{x} =0$
$x= \pm \sqrt 2$

Question 16
Prove that
if $0 < x < \frac {\pi}{2}$
$cot^{-} [ \frac {\sqrt {1 -sin x} + \sqrt {1 +sin x}}{\sqrt {1 -sin x} - \sqrt {1 +sin x}}] = \pi - \frac {x}{2}$

Question 17
Find the value
$\sum_{k=1}^{n} sin^{-1} \frac {\sqrt k - \sqrt {k-1}}{\sqrt {k(k+1)}}$
$tan^{-1} \sqrt n$

Question 18
Find the set of values of x for which $2tan^{-1}x + cos^{-1} \frac {1-x^2}{1+x^2}$ is independent of x
$(-\infty, 0]$

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