Inverse Trigonometric Functions — JEE Main PYQs

If $\frac{\pi}{4} + \displaystyle\sum_{p=1}^{11} \tan^{-1}\left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to __________.

Let $0<\alpha<1$, $\beta=\frac{1}{3\alpha}$ and $\tan^{-1}\left(1-\alpha\right)+\tan^{-1}\left(1-\beta\right)=\frac{\pi}{4}$. Then $6(\alpha+\beta)$ is equal to:

If $\sin\left(\tan^{-1}\left(x\sqrt{2}\right)\right) = \cot\left(\sin^{-1}\sqrt{1 - x^2}\right)$, $x \in (0, 1)$, then the value of $x$ is :

Let $\alpha = 3\sin^{-1}\left(\frac{6}{11}\right)$ and $\beta = 3\cos^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values.

Given below are two statements :

Statement I : $\cos(\alpha + \beta) > 0$.

Statement II : $\cos(\alpha) < 0$.

In the light of the above statements, choose the correct answer from the options given below :

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

Considering the principal values of the inverse trigonometric functions, $\sin^{-1}\left(\frac{\sqrt{3}}{2}x + \frac{1}{2}\sqrt{1 - x^2}\right)$, $-\frac{1}{2} < x < \frac{1}{\sqrt{2}}$, is equal to

The sum of the infinite series $\cot^{-1}\left(\dfrac{7}{4}\right)+\cot^{-1}\left(\dfrac{19}{4}\right)+\cot^{-1}\left(\dfrac{39}{4}\right)+\cot^{-1}\left(\dfrac{67}{4}\right)+\ldots$ is :

The value of $\cot^{-1}\left(\frac{\sqrt{1+\tan^2(2)}-1}{\tan(2)}\right) - \cot^{-1}\left(\frac{\sqrt{1+\tan^2\left(\frac{1}{2}\right)}+1}{\tan\left(\frac{1}{2}\right)}\right)$ is equal to

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is :

If $\frac{\pi}{2} \le x \le \frac{3\pi}{4}$, then $\cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right)$ is equal to

If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1}\alpha) + \operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2 + \beta$ is _____.

If $\alpha > \beta > \gamma > 0$, then the expression

$\cot^{-1}\left\{\beta + \dfrac{(1 + \beta^2)}{(\alpha - \beta)}\right\} + \cot^{-1}\left\{\gamma + \dfrac{(1 + \gamma^2)}{(\beta - \gamma)}\right\} + \cot^{-1}\left\{\alpha + \dfrac{(1 + \alpha^2)}{(\gamma - \alpha)}\right\}$ is equal to :

$\cos\left(\sin^{-1}\frac{3}{5} + \sin^{-1}\frac{5}{13} + \sin^{-1}\frac{33}{65}\right)$ is equal to:

Given that the inverse trigonometric function assumes principal values only. Let $x$, $y$ be any two real numbers in $[-1, 1]$ such that $\cos^{-1} x - \sin^{-1} y = \alpha$, $\dfrac{-\pi}{2} \le \alpha \le \pi$. Then, the minimum value of $x^2 + y^2 + 2xy \sin\alpha$ is

For $n \in \mathbb{N}$, if $\cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \dfrac{\pi}{4}$, then $n$ is equal to ________.

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is ________.

For $\alpha, \beta, \gamma \neq 0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^{2}+b^{2}$ is equal to