Basic Inverse Trigonometric Functions

Q.

Prove that $\cos \left[\left(\frac{3\mathrm{\pi }}{4}\right)+x\right]-\cos \left[\left(\frac{3\mathrm{\pi }}{4}\right)-x\right]=-\surd 2\sin x.$

Q. The value of ${\tan }^{-1}\frac{4}{7}+{\tan }^{-1}\frac{4}{19}+{\tan }^{-1}\frac{4}{39}+{\tan }^{-1}\frac{4}{67}+\cdots \infty$ equals

1. ${\tan }^{-1}1+{\tan }^{-1}\frac{1}{2}$
2. $\frac{\pi }{2}-{\cot }^{-1}2$
3. $\frac{\pi }{2}-{\cot }^{-1}1$
4. ${\cot }^{-1}1+{\tan }^{-1}3$

Q. If a function is invertible, then the function must be

1. One-to-One Function
2. Many - to - One function
3. On to function
4. Bijection

Q.

Solve: $se{c}^{2}45°+co{t}^{2}135°$

Q. f(x) = cos(x) defined on f : [a, b] $\to$ [-1, 1] has its inverse then which of the following is correct?

1. $a=-\pi ;b=\pi$
2. $a=0;b=\pi$
3. $a=0;b=2\pi$
4. None of these

Q. Find the value of the expression
$3{\sec }^{-1}\left(2\right)-6{\sin }^{-1}\left(\frac{1}{2}\right).$

1. $2\pi$
2. $\frac{5\pi }{6}$
3. $0$
4. $1$

Q. $y=si{n}^{-1}\left(x\right)$ is the inverse of the function y=sin(x).

1. True
2. False

Q. A solution of the equation ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\pi }{2}$ is

1. $x=\pi$
2. $x=0$
3. $x=1$
4. $x=-1$

Q. Find the value of
$co{t}^{-1}\frac{6}{8}+{\sin }^{-1}\frac{5}{13}.$

1. ${\cos }^{-1}\frac{16}{65}$
2. ${\sin }^{-1}\frac{63}{65}$
3. ${\sin }^{-1}\frac{126}{130}$
4. None

Q. If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)$ and $\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is/are

1. $\cos \beta >0$
2. $\sin \beta <0$
3. $\cos (\alpha +\beta )>0$
4. $\cos \alpha <0$

Q. Statement $1$: The value of ${\cos }^{-1}\left(\cos 30\right)$ is $30-9\pi$.

Statement $2:30-9\pi \in [0,\pi ]$.

1. STATEMENT $1$ is TRUE and STATEMENT $2$ is FALSE.
2. Both the statements are TRUE, and STATEMENT $2$ is NOT the correct explanation of STATEMENT $1$.
3. Both the statements are TRUE, and STATEMENT $2$ is the correct explanation of STATEMENT $1$.
4. STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.

Q. If ${\cos }^{-1}\left(\frac{2}{3x}\right)+{\cos }^{-1}\left(\frac{3}{4x}\right)=\frac{\pi }{2}(x>\frac{3}{4}),$ then $x$ is equal to:

1. $\frac{\sqrt{145}}{12}$
2. $\frac{\sqrt{146}}{12}$
3. $\frac{\sqrt{145}}{11}$
4. $\frac{\sqrt{145}}{10}$

Q. If $\frac{\tan 2^{\circ }\cdot \tan {2017}^{\circ }\cdot \tan {2019}^{\circ }}{\tan {2019}^{\circ }-\tan {2017}^{\circ }-\tan 2^{\circ }}=a$, then the value of ${\tan }^{-1}a$(in degrees) is

Q. $2\pi -({\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{16}{65})$ is equal to:

1. $\frac{5\pi }{4}$
2. $\frac{\pi }{2}$
3. $\frac{3\pi }{2}$
4. $\frac{7\pi }{4}$

Q.

$sin\left(2si{n}^{-1}\sqrt{\frac{63}{65}}\right)=$

1. $\frac{2\sqrt{126}}{65}$
   

2. $\frac{4\sqrt{63}}{65}$
   

3. $\frac{8\sqrt{63}}{65}$
   

4. $\frac{\sqrt{65}}{63}$
   

Q. If ${\sin }^{-1}x+{\cot }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{2},$ then the value of $x$ is

1. $0$
2. $\frac{2}{\sqrt{5}}$
3. $\frac{1}{\sqrt{5}}$
4. $\frac{\sqrt{3}}{2}$

Q. If $\sin \alpha =\frac{-4}{5}$, $\alpha \in \left[\frac{3\pi }{2},2\pi \right]$, then the value of $\cos \frac{\alpha }{2}$ is equal to

यदि $\sin \alpha =\frac{-4}{5}$, $\alpha \in \left[\frac{3\pi }{2},2\pi \right]$, तब $\cos \frac{\alpha }{2}$ का मान बराबर है

1. $\frac{-1}{\sqrt{5}}$
2. $\frac{1}{\sqrt{5}}$
3. $\frac{2}{\sqrt{5}}$
4. $\frac{-2}{\sqrt{5}}$

Q. Domain of the function $f\left(x\right)={\tan }^{-1}\sqrt{1-\frac{1}{x}}$ is

फलन $f\left(x\right)={\tan }^{-1}\sqrt{1-\frac{1}{x}}$ का प्रांत है

1. R⁺
2. [0, 1)
3. (–∞, 0) ∪ [1, ∞)
4. (–1, ∞)

Q.

$se{c}^{-1}\left[sec\right(-{30}^{\circ }\left)\right]=$
[MP PET 1992]

1. $-{60}^{\circ }$
2. $-{30}^{\circ }$

3. ${30}^{\circ }$

4. ${150}^{\circ }$

Q. The value of ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$ is equal to

${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$ का मान है

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{3}$
3. $-\frac{\pi }{3}$
4. $-\frac{\pi }{6}$

Q. Find the value of the expression
$3{\sec }^{-1}\left(2\right)-6{\sin }^{-1}\left(\frac{1}{2}\right).$

1. $2\pi$
2. $\frac{5\pi }{6}$
3. $0$
4. $1$

Q. The value of is $si{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)-si{n}^{-1}\left(\frac{1}{2}\right)$ is

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${15}^{\circ }$
4. ${30}^{\circ }$

Q. The value of x (x > 0) for which $\tan \left({\sec }^{-1}\frac{1}{x}\right)=\sin \left({\tan }^{-1}2\right)$ is

x (x > 0) के किस मान के लिए $\tan \left({\sec }^{-1}\frac{1}{x}\right)=\sin \left({\tan }^{-1}2\right)$ है?

1. $\frac{\sqrt{5}}{3}$
2. $\frac{1}{2}$
3. $\frac{2}{\sqrt{5}}$
4. $\frac{3}{\sqrt{5}}$

Q. Find the value of
$co{t}^{-1}\frac{6}{8}+{\sin }^{-1}\frac{5}{13}.$

1. ${\cos }^{-1}\frac{16}{65}$
2. ${\sin }^{-1}\frac{63}{65}$
3. ${\sin }^{-1}\frac{126}{130}$
4. None

Q. The value of ${\tan }^{-1}\frac{4}{7}+{\tan }^{-1}\frac{4}{19}+{\tan }^{-1}\frac{4}{39}+{\tan }^{-1}\frac{4}{67}+\cdots \infty$ equals

1. ${\tan }^{-1}1+{\tan }^{-1}\frac{1}{2}$
2. $\frac{\pi }{2}-{\cot }^{-1}2$
3. $\frac{\pi }{2}-{\cot }^{-1}1$
4. ${\cot }^{-1}1+{\tan }^{-1}3$

Q. If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$, then ${\cos }^{-1}x+{\cos }^{-1}y$ is equal to

1. $\frac{3\pi }{4}$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q. Find the value of
$ta{n}^2\left(se{c}^{-1}\left(3\right)\right)$+$co{t}^2\left(cose{c}^{-1}\left(4\right)\right)$

1. $5$
2. $25$
3. $7$
4. $23$

Q. $\int \frac{1}{1+\sin x}dx$ is equal to
(where $C$ is constant of integration)

1. $\tan x+{\sec }^{2}x+C$
2. $\tan x-\sec x+C$
3. ${\tan }^{2}x+\sec x+C$
4. $\tan x+\sec x+C$

Q. $co{s}^{-1}\left(\frac{15}{7}\right)+2ta{n}^{-1}\left(\frac{1}{5}\right)=$

1. $\frac{\pi }{2}$
   
2. $co{s}^{-1}\left(\frac{171}{221}\right)$
   
3. $\frac{\pi }{4}$
4. None of these

Q. The value of $si{n}^{-1}\left\{cot(si{n}^{-1}\sqrt{\left(\frac{2-\sqrt{3}}{4}\right)}+co{s}^{-1}\frac{\sqrt{12}}{4}+se{c}^{-1}\sqrt{2})\right\}$ is:

1. $0$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{2}$