Inverse Trigonometric Functions - Trigonometry - Mathematics Previous Year Questions

Consider the following statements

Assertion (A) For $x \in R-\{1\}$;

$$\frac{d}{d x}\left(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\right)=\frac{d}{d x}\left(\tan ^{-1} x\right)$$

Reason (R) For $x<1, \tan ^{-1}\left(\frac{1+x}{1-x}\right)=\frac{\pi}{4}+\tan ^{-1} x$, for

$$x>1, \tan ^{-1}\left(\frac{1+x}{1-x}\right)=-\frac{3 \pi}{4}+\tan ^{-1} x$$

The correct answer is

If $y=\left(\sin ^{-1} x\right)^2$, then $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}=$

Consider the following statements

Assertion (A) : When $x, y, z$ are positive numbers, then

$$\begin{aligned} & \tan ^{-1}\left(\sqrt{\frac{x(x+y+z)}{y z}}\right)+\tan ^{-1}\left(\sqrt{\frac{y(x+y+z)}{x z}}\right) +\tan ^{-1}\left(\sqrt{\frac{z(x+y+z)}{x y}}\right)=\pi \end{aligned}$$

Reason (R) : $\tan ^{-1} a+\tan ^{-1} b=\tan ^{-1}\left(\frac{a+b}{1-a b}\right)$, if $a>0$ and $b>0$

The correct answer is

If $e^{\left(\sinh ^{-1} 2+\cosh ^{-1} \sqrt{6}\right)}=(a+(b+\sqrt{c}) \sqrt{a}+b \sqrt{c})$, then $a+b+c=$

If $2 \tanh ^{-1} x=\sinh ^{-1}\left(\frac{4}{3}\right)$, then $\cosh ^{-1}\left(\frac{1}{x}\right)=$

$$\tan ^{-1} \frac{3}{5}+\tan ^{-1} \frac{6}{41}+\tan ^{-1} \frac{9}{191}=$$

The range of the real value function $f(x)=\sin ^{-1}\left(\sqrt{x^2+x+1}\right)$ is

If $f(x)=\sqrt{\cos ^{-1} \sqrt{1-x^2}}$, then $f^{\prime}\left(\frac{1}{2}\right)=$

The number of values of $x$ satisfying the equation, $\tan ^{-1}\left(x+\frac{\sqrt{2}}{x}\right)+\tan ^{-1}\left(x-\frac{\sqrt{2}}{x}\right)=\tan ^{-1}(x)$ is

If $y=\sec ^{-1} x$, then $\frac{d^2 y}{d x^2}=$