Inverse Trigonometry Questions - JEE

$\tan^{-1} x + \tan^{-1}\dfrac{2x}{1-3x^2} = \pi + \tan^{-1}\dfrac{3x - x^3}{1-3x^2}$ $(x > 0)$ is true if

$\tan^{-1}\!\left(1-x^2-\dfrac{1}{x^2}\right)+\sin^{-1}\!\left(x^2+\dfrac{1}{x^2}-1\right)$, $x\ne 0$, equals:

262. If $\sec^{-1}(x) + \tan^{-1}\sqrt{9y^2 - 1} + \sin^{-1}(x^2 + y^2) = \lambda$ has no solution, then exhaustive set of values of $\lambda$ is equal to:

Let $f: R \to \left(0, \dfrac{2\pi}{3}\right]$ defined as $f(x) = \cot^{-1}(x^2 - 4x + \alpha)$. The smallest integral value of $\alpha$ such that $f(x)$ is an into function, is equal to:

Let $f(x) = \sin x + \cos x + \tan x + \arcsin x + \arccos x + \arctan x$. If $M$ and $m$ are maximum and minimum values of $f(x)$, then their arithmetic mean is equal to

239. Let $a \in \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$ such that $\tan^{-1}\!\left(\dfrac{\tan\alpha}{3 + 2\tan^2\alpha}\right) + \tan^{-1}\!\left(\dfrac{2\tan\alpha}{3}\right) = \dfrac{\pi}{12}$, then $\alpha$ equals:

$4\cot^{-1} 3 + \sin^{-1}\dfrac{1}{\sqrt{5}} - \sin^{-1}\dfrac{1}{\sqrt{5}} = \underline{\quad}$.

142. If $x=\sin^{-1}(\sin 10)$ and $y=\cos^{-1}(\cos 10)$, then $y-x$ is equal to:

934. Find the number of integers not in the domain of $f(x) = \cos^{-1}\!\left(\dfrac{2 - x}{2x}\right)$.

Find the value of m where m = tan²(sec⁻¹ 2) + cot²(cosec⁻¹ 3), then find m² + m + 10

If $\frac{x + \frac{3}{x}}{} = 2$, then the value of $\sin^{-1} x$ is

Match relations on $A=\{a,b,c\}$: (I)–(IV) with properties P,Q,R,S.

If \[2y = \cot^{-1}\left(\frac{3\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)^2\], where $x \in \left(0, \frac{\pi}{2}\right)$, then $\frac{dy}{dx}$ is equal to

$\sin^{-1}(\sin 10)=$

If $\cos^{-1}x-\cos^{-1}(y/2)=\alpha$, then $4x^2-4xy\cos\alpha+y^2=$

Sum to infinite terms of the series $\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{2}{2^3} + \tan^{-1}\frac{2}{2^5} + \cdots + \tan^{-1}\frac{1}{2^{2n-1}} + \cdots$ is

$3\tan^{-1} x = \tan^{-1}\dfrac{3x - x^3}{1-3x^2}$ $(x > 0)$ if \(x

If the sum of all the solutions of \tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}, \quad -1 < x < 1, x \neq 0, \text{ is } \alpha - \frac{4}{\sqrt{3}}, \text{ then } \alpha \text{ is equal to}

Solve $\sin[2\cos^{-1}\{\cot(2\tan^{-1}x)\}] = 0$; express $x$ in the form $a \pm b$.

For $ x \in (-\infty, -1] $, if $ f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) - 2\tan^{-1}x $ and $ g(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 4\tan^{-1}x $, then $ f(x) + g(x) $ equals:

If $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\pi/2$, then $xy+yz+zx=$

The domain of $f(x)=\sin^{-1}(\log_2(x/3))$ is:

Let $f(x)=\sin^{-1}(\tan x)+\cos^{-1}(\cot x)$. Which are true?

The solution set of the equation $\sin^{-1}\sqrt{1-x^2} + \cos^{-1} x = \cot^{-1}\frac{\sqrt{1-\sin^2 x}}{\sin x}$

$\tan^{-1}\dfrac{1}{2} + 2\tan^{-1}\dfrac{1}{5} + \sin^{-1}(\underline{\quad})$ (if \(x

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

The value of $\displaystyle\sum_{\omega=1}^{\infty} \sin^{-1}\!\left[\dfrac{2\omega+1}{\omega(\omega+1)(\sqrt{\omega^2+2\omega}+\sqrt{\omega^2-1})}\right]$ is equal to:

If $\tan\left[\tan^{-1}2 + \tan^{-1}20k\right] = k$, then the sum of all solutions $k_1 + k_2$ equals?

Let $ y = \tan^{-1}\!\left(\dfrac{4x}{1+5x^2}\right) + \tan^{-1}\!\left(\dfrac{2+3x}{3-2x}\right) $ where $ x \in \left(0, \dfrac{2}{3}\right) $. If $ \dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2} $, then the value of $ \alpha $ is equal to:

The numerical value of $\tan^{-1}\dfrac{1}{2} + \tan^{-1}\dfrac{1}{3}$ is ______.

968. The number of real solutions of the equation $\sqrt{1+\cos 2x} = \sqrt{2}\sin^{-1}(\sin x)$ where $-\pi \leq x \leq \pi$.

$\displaystyle\sum_{n=1}^{\infty}\tan^{-1}\!\frac{1}{2n^2}=$

Consider $f(x) = \dfrac{|x - 4|}{|x| + 1}$. If sum of all distinct possible values of $\sin^{-1}(\sin[f(x)])$ is $a\pi + b$ then find the absolute value of $(a + b)$.[Note: $[z]$ denotes greatest integer function less than or equal to $z$.]

$\cot^{-1}\!\left(\dfrac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right)$, \(\pi/2

Range of $f(x)=\sin^{-1}(\log_2(-x^2+2x+3))$ is:

If $\cos^{-1}\!\sqrt{p}+\cos^{-1}\!\sqrt{1-p}+\cos^{-1}\!\sqrt{1-q}=\frac{3\pi}{4}$, then $q=$

Number of solutions of $\tan^{-1}(2x)+\tan^{-1}(3x)=\pi/4$ for $x\in(0,1)$:

Numerical value of $\tan[2\tan^{-1}(1/5)-\pi/4]$ is:

968. The number of real solutions of the equation $\sqrt{1+\cos 2x} = \sqrt{2}\sin^{-1}(\sin x)$ where $-\pi \leq x \leq \pi$.

Let $ y = \tan^{-1}\!\left(\dfrac{4x}{1+5x^2}\right) + \tan^{-1}\!\left(\dfrac{2+3x}{3-2x}\right) $ where $ x \in \left(0, \dfrac{2}{3}\right) $. If $ \dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2} $, then the value of $ \alpha $ is equal to:

If $y = \cos^{-1}\!\cos\!\left(\log_2 2^{\ln e^{\sin^{-1}\sin x}}\right)$ for $-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$, then $y$ equals:

If $\sum_{r=1}^{100} \sin^{-1}\left(\dfrac{1}{\sqrt{r^2+1}\sqrt{r^2+2r+2}}\right)$ is equal to $\tan^{-1}\left(\dfrac{p}{q}\right)$ where $p$ and $q$ are co-prime, then the value of $(p+q)$ is equal to:

$\sin^{-1}(\sin 10)=$

If $\sum_{i=1}^{10}\sin^{-1}x_i=5\pi$, then $\sum_{i=1}^{10}x_i^2=$

$\cos[\tan^{-1}\{\sin(\cot^{-1}x)\}]=$

If $2\le a, value of \(\cos^{-1}[a]+\text{cosec}^{-1}[a]+\cot^{-1}[a]$ (where $[\cdot]$ is GIF):

If $\cos^{-1}x-\cos^{-1}(y/2)=\alpha$, then $4x^2-4xy\cos\alpha+y^2=$

$\tan\!\left[\frac{\pi}{4}+\frac{1}{2}\cos^{-1}x\right]+\tan\!\left[\frac{\pi}{4}-\frac{1}{2}\cos^{-1}x\right]$ equals:

If $\alpha=2\tan^{-1}\!\frac{1+x}{1-x}$ and $\beta=\sin^{-1}\!\frac{1-x^2}{1+x^2}$ for $x>1$, then $\alpha+\beta=$

$\displaystyle\sum_{r=0}^{\infty}\tan^{-1}\!\frac{1}{r^2+3r+3}=$

Inverse Trigonometry Questions (1043)