Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions

Q. ${\cos }^{-1}\left(\cos \right(-5\left)\right)+{\sin }^{-1}\left(\sin \right(6\left)\right)-{\tan }^{-1}\left(\tan \right(12\left)\right)$ is equal to
(The inverse trigonometric functions take the principal values)

1. $3\pi +1$
2. $3\pi -11$
3. $4\pi -11$
4. $4\pi -9$

Q. The value of ${\sin }^{-1}\left(\frac{12}{13}\right)-{\sin }^{-1}\left(\frac{3}{5}\right)$ is equal to :

1. $\frac{\pi }{2}-{\cos }^{-1}\left(\frac{9}{65}\right)$
2. $\pi -{\sin }^{-1}\left(\frac{63}{65}\right)$
3. $\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$
4. $\pi -{\cos }^{-1}\left(\frac{33}{65}\right)$

Q.

The smallest and the largest values of ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$ , $0\le x\le 1$ are

1. $0,\mathrm{\pi }$

2. $0,\frac{\mathrm{\pi }}{4}$

3. $\frac{-\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4}$

4. $\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}$

Q.

$1-{\cos }^{2}A$ is equal to ?

Q.

If $0<x<\pi$ and $\cos \left(x\right)+\sin \left(x\right)=\frac{1}{2}$, then $\tan \left(x\right)$ is equal to

1. $\frac{\left(4-\sqrt{7}\right)}{3}$

2. $-\frac{\left(4+\sqrt{7}\right)}{3}$

3. $\frac{\left(1+\sqrt{7}\right)}{4}$

4. $\frac{\left(1-\sqrt{7}\right)}{4}$

Q. For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $[0,\pi ],$ which of the following options is/are correct?

1. $f\left(4\right)=\frac{\sqrt{3}}{2}$
2. $\underset{n\to \infty }{lim}f\left(n\right)=\frac{1}{2}$
3. $\text{If}\alpha =\tan \left({\cos }^{-1}f\right(6\left)\right),\text{then}{\alpha }^2+2\alpha -1=0$
4. $\sin \left(7{\cos }^{-1}f\right(5\left)\right)=0$

Q.

If tanA+1/tanA=2 show that tan^{​​​​ 2} A +1/tan^{​​​​ 2} A=2

Q. The value of ${\tan }^{-1}\left(\cot \frac{43\pi }{4}\right)$ is

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

Q.

If $f\left(x\right)=\tan x-{\tan }^{3}x+{\tan }^{5}x-.........\infty$ with $0<x<\frac{\mathrm{\pi }}{4}$, then ${\int }_0^{\frac{\mathrm{\pi }}{4}}f\left(x\right)dx=$

1. $1$

2. $0$

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

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

5. $-\frac{1}{4}$

Q. Find the value of the trigonometric function $\sin (-\frac{11\pi }{3})$.

Q.

If ${\left({\tan }^{-1}x\right)}^2+{\left({\cot }^{-1}x\right)}^2=\frac{5{\pi }^2}{8}$, then $x=$

1. $0$

2. $2$

3. $1$

4. $-1$

5. $2\sqrt{2}$

Q.

If ${\tan }^{-1}\left(\tan \left(\frac{5\pi }{4}\right)\right)=\alpha$ and ${\tan }^{-1}\left(-\tan \left(\frac{2\pi }{3}\right)\right)=\beta$ then

1. $4\alpha -4\beta =0$

2. $4\alpha -3\beta =0$

3. $\alpha >\beta$

4. None of these

Q.

If k is a scalar and I is a unit matrix of order 3, then adj(kI) = [MP PET 1991;Pb. CET 2003]

1. 

2. 

3. 

4. 

Q.

Solution of the equation tan ^-1 (x - 1) + tan ^-1 x + tan ^-1 (x+1)=tan ^-1 3x is

1. 
   

2. 
3. x = 0
4. None of these

Q. If $\sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=1$, then find the value of $x$.

Q. If $L={\sin }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)$ and $M={\cos }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right),$ then:

1. $M=\frac{1}{2\sqrt{2}}+\frac{1}{2}\cos \frac{\pi }{8}$
2. $M=\frac{1}{4\sqrt{2}}+\frac{1}{4}\cos \frac{\pi }{8}$
3. $L=-\frac{1}{2\sqrt{2}}+\frac{1}{2}\cos \frac{\pi }{8}$
4. $L=\frac{1}{4\sqrt{2}}-\frac{1}{4}\cos \frac{\pi }{8}$

Q. Number of solutions of the equation $2e^{|x|}{\tan }^{-1}|x|=1$ is

Q. If the value of $\int (\sin 3x\cdot \cos 3x\cdot \cos 6x\cdot \cos 12x)dx=\frac{\cos ax}{b}+C,$ then the absolute value of $\frac{b}{a}$ is
(where $C$ is constant of integration)

Q. Prove that ${\sin }^2\frac{\pi }{6}+{\cos }^2\frac{\pi }{3}-{\tan }^2\frac{\pi }{4}=-\frac{1}{2}$

Q.

cot^{​​​​-1} (x) + tan^{​​​​-1} (3) = π/2 , then x = ?

Q. Prove that: ${\cot }^2\frac{\pi }{6}+\text{cosec}\frac{5\pi }{6}+3{\tan }^2\frac{\pi }{6}=6$

Q. The value of $\tan \frac{\pi }{8}$ is equal to

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

Q. The value of $\int \frac{\sqrt{x^2-a^2}}{x}dx$ will be

1. $\sqrt{(x^2-a^2)}-ata{n}^{-1}\left[\frac{\sqrt{(x^2-a^2)}}{a}\right]$
2. $\sqrt{(x^2-a^2)}+ata{n}^{-1}\left[\frac{\sqrt{(x^2-a^2)}}{a}\right]$
3. $\sqrt{(x^2-a^2)}+a^{2}ta{n}^{-1}\left[\sqrt{x^2-a^2}\right]$
4. $\frac{ta{n}^{-1}x}{a+c}$

Q. The number of common solution(s) of $y=\sin x$ and $y=(2x-\pi {)}^2$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. Find the value of $\tan \frac{13\pi }{12}$

Q. If $y={\tan }^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$ ,
then $\frac{dy}{dx}$ is equal to

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

Q. $\sin \{{\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\frac{1}{2}\}=$

1. $0$
2. $-1$
3. $2$
4. $1$

Q.

Evaluate: $\cos \left(\frac{\mathrm{\pi }}{3}\right)$

Q. If $({\tan }^{-1}x{)}^2+({\cot }^{-1}x{)}^2=\frac{5{\pi }^2}{8},$ then the value of $x$ is

1. $0$
2. $-1$
3. $-2$
4. $-3$

Q. If $y=\log \sqrt{\frac{1+\tan x}{1-\tan x}}$, prove that $\frac{dy}{dx}=\sec 2x$.