Trigonometry Questions - JEE

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 $\alpha+\beta=$

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

Range of $f(x)=\cot^{-1}(\log_e(1-x^2))$ is:

Range of $f(x)=\sin^{-1}x+\tan^{-1}x+\sec^{-1}x$ is:

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

$\sin^{-1}(\sin 5)>x^2-4x$ holds for:

If $\sin(2\cos^{-1}\frac{1}{\sqrt{5}})+\cos(2\tan^{-1}\frac{1}{3})=\frac{p}{q}$ (coprime), units digit of $(p-q)^{2k+1}$, $k\in\mathbb{N}$ can be:

If $\sin^{-1}\!\sqrt{x/2}+\sin^{-1}\!\sqrt{1-x/4}+\tan^{-1}y=\frac{2\pi}{3}$, then which are true?

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

For $\alpha,\beta,\gamma\neq0$. 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 we consider only the principal values of the inverse trigonometric functions, then the value of $\tan^{-1}\left(\cos^{-1}\frac{1}{5} - \sin^{-1}\frac{4}{17}\right)$ is

The complete set of values of $x$ satisfying the inequality $\sin^{-1}(\sin 5) > x^2 - 4x$ is $(2 - \sqrt{\lambda - 2\pi}, 2 + \sqrt{\lambda - 2\pi})$, then $\lambda =$

If $\sin^{-1}a + \sin^{-1}b + \sin^{-1}c = \pi$, then the value of $a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2}$ will be

Ex. 34. Statement I: If α, β are roots of 6x² + 11x + 3 = 0, then cos α exists but not cos⁻¹β (α > 0).Statement II: Domain of cos⁻¹x is [−1, 1].

$\sin^{-1}x > \cos^{-1}x$ holds for

Let $\cos^{-1}(4x^3 - 3x) = a + b\cos^{-1}x$.If $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$, then $\sin^{-1}\left(\sin\frac{a}{b}\right)$ is:

Trigonometry Questions (1127)