Inverse Trigonometry Questions (1043)

If $y = \tan^{-1}\dfrac{4x}{1+5x^2} + \tan^{-1}\dfrac{2+3x}{3-2x}$, find $\dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2}$. Find $\alpha$.

If \sin\theta = 3\sin(\theta + 2\alpha)\, then the value of \tan(\theta + \alpha) + 2\tan\alpha\ is

Let \(f(x) = 1 + 2\sin\left(\frac{\pi x}{e^x+1}\right)\), \(x > 0\), then \(f^{-1}(x)\) is equal to (assuming \(f\) is bijective)

Let \(g: \mathbb{R} \to \left[0, \frac{7\pi}{2}\right)\) is defined by \(g(x) = \cos^{-1}\frac{x}{1+x^2}\). Then the possible values of \(k\) for which \(g\) is a surjective function, is

Find the number of solutions to the equation \(y = |x^2 - 1| = |\tan^{-1}|x||\)

If \(\alpha = \frac{1}{3}\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{1}{3}\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) where \(x \geq \frac{4}{3}\), then the value of \(\dfrac{\cos 2\alpha + \sec\alpha + 3\sqrt{3}}{\sqrt{3}}\) is equal to:

If f(x) = ∑r=1n [tan−1(x+r) − tan−1(x+r−1)], then limx→0 f'(x) is

The value of x satisfying (cot−1x)(tan−1x) + 2(\(\frac{π}{2}\) − cot−1x) − 3tan−1x − 3(\(\frac{π}{2}\)) ≥ 0 is

Let set $A$ denote the solutions of $\cos^{-1}(4x^3-3x)=\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$. Then

Let $\frac{5}{6}\cos^{-1}\sqrt{\dfrac{3}{3+\pi^2}}+\frac{1}{3}\sin^{-1}\dfrac{2\sqrt{3}\pi}{3+\pi^2}+\frac{1}{6}\tan^{-1}\dfrac{\sqrt{3}}{\pi}=a$ and $\cos^{-1}\!\left[\frac{13}{40}\cos\!\left(\cot^{-1}\frac{5}{12}\right)+\frac{13}{32}\sin\!\left(\cos^{-1}\frac{5}{13}\right)\right]=b$. Then $\csc\!\left(\displaystyle\int_b^a\left[\frac{\tan x}{\sqrt{3}}\right]dx\right)$ is ($[\cdot]$ = GIF)

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in meters) above the line AC is (JEE Main 2020)

If \(\dfrac{\cos x + \cos y + \cos z}{\cos(x+y+z)} = 2\) and \(\dfrac{\sin x + \sin y + \sin z}{\sin(x+y+z)} = 2\), then the value of \(\cos(x+y) + \cos(y+z) + \cos(z+x)\) is equal to: (where \(x, y, z \in R\))

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

Considering only principal values of inverse trigonometric functions, the value of $\tan\!\left(\sin^{-1}\frac{3}{5}-2\cos^{-1}\frac{2}{5}\right)$ is:

$\displaystyle\sum_{n=1}^\infty\cot^{-1}\!\left(\frac{(2n^2+2n+1)(n^2+n+1)}{n^4+2n^3+2n^2+2n+2}\right)=\sec^{-1}\!\left(\frac{5}{\lambda}\right)$. Then $\lambda$ equals:

Considering only principal values of inverse trigonometric functions, the value of $\tan\!\left(\sin^{-1}\frac{3}{5}-2\cos^{-1}\frac{2}{5}\right)$ is:

$\displaystyle\sum_{n=1}^\infty\cot^{-1}\!\left(\frac{(2n^2+2n+1)(n^2+n+1)}{n^4+2n^3+2n^2+2n+2}\right)=\sec^{-1}\!\left(\frac{5}{\lambda}\right)$. Then $\lambda$ equals:

If the solution of the equation $\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, is $\sin^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha$, $\beta$ are integers, then $\alpha + \beta$ is equal to:

The number of solutions of the equation: $x^2 + (x+1)\sin\frac{\pi x}{6} = \frac{3+x}{2}$; $-2 \leq x \leq 0$

The period of the function $f(x) = e^{\sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right)}$ is:

If $u=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}+\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}$, then the difference between maximum and minimum values of $u^2$ is given by:

A quadrilateral $ABCD$ in which $AB = a$, $BC = b$, $CD = c$ and $DA = d$ is such that one circle can be inscribed in it and another circle can be circumscribed about it. $\cos A =$

The number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\dfrac{\pi}{6}$, where $-\dfrac{1}{2\sqrt{6}}<x<\dfrac{1}{2\sqrt{6}}$, is equal to

If the angles of elevation of the top of a tower from three collinear points $A$, $B$ and $C$ on a line leading to the foot of the tower are $30°$, $45°$ and $60°$ respectively, then the ratio $AB : BC$ is

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

If $[\sin^{-1}\cos x - \sin^{-1}\tan x - 1] = 1$ whose $[.]$ denotes the greatest integer function, then $x$ belongs to:

Roots \(r,s,t\) of \(x(x-2)(3x-7)=2\) are real and positive. \(\tan^{-1}r+\tan^{-1}s+\tan^{-1}t=\)

If $\dfrac{\tan(A-B)}{\tan A}+\dfrac{\sin^2C}{\sin^2A}=1$, $A,B,C\in\left(0,\dfrac{\pi}{2}\right)$, then

Two rays are drawn through a point $A$ at an angle of $30°$. A point $B$ is taken on one of them at a distance $a$ from the point $A$. A perpendicular is drawn from the point $B$ to the other ray and another perpendicular is drawn from its foot to $AB$ to meet $AB$ at another point from where the similar process is repeated indefinitely. The length of the resulting infinite polygon line is:

Let $\dfrac{\pi}{2}<\theta<\pi$ and $\cot\theta=-\dfrac{1}{2\sqrt{2}}$. Then the value of $\sin\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta+\sin8\theta)+\cos\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta-\sin8\theta)$ is equal to

The least value of $\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}$ is: (Where $A, B, C$ are interior angles of a triangle)

If the equation $a_1 + a_2 \cos 2x + a_3 \sin^2 x = 1$ is satisfied by every real value of $x$, then the number of possible values of the triplet $(a_1, a_2, a_3)$ is:

For a triangle ABC, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

If $\pi < \theta < \frac{3\pi}{2}$ and $\cos \theta = -\frac{3}{5}$, then $\tan \left(\frac{\theta}{2}\right)$ is equal to

A tower of height 50 m is located on top of a hill opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_1$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is

Let S = \left\{x \in \mathbb{R} : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}. If n(S) denotes the number of elements in S then:

Let S be the set of all solutions of the equation \cos^{-1}(2x) - 2\cos^{-1}(\sqrt{1-x^2}) = \pi, \quad x \in \left[-\frac{1}{2}, \frac{1}{2}\right]. Then \sum_{x \in S} 2\sin^{-1}(x^2 - 1) \text{ is equal to}

Let $\theta \in \left(0, \frac{\pi}{4}\right)$ and $t_1 = (\tan \theta)^{\tan \theta}$, $t_2 = (\tan \theta)^{\cot \theta}$, $t_3 = (\cot \theta)^{\tan \theta}$ and $t_4 = (\cot \theta)^{\cot \theta}$, then:

In a $\triangle ABC$, $\angle B=\frac{\pi}{3}$ and $\angle C=\frac{\pi}{4}$, also $D$ divides $BC$ internally in the ratio $1:3$, then $\frac{\sin\angle BAD}{\sin\angle CAD}$ is equal to:

\(\text{cosec}^{-1}(\cos x)\) exists if:

If $\cot x=\dfrac{5}{12}$ for some $x\in\left(\pi,\dfrac{3\pi}{2}\right)$, then $\sin7x\left(\cos\dfrac{13x}{2}+\sin\dfrac{13x}{2}\right)+\cos7x\left(\cos\dfrac{13x}{2}-\sin\dfrac{13x}{2}\right)$ is equal to

If \(\alpha,\beta\) are roots of \(x^2-3x+2=0\), then \(\tan^{-1}\alpha+\tan^{-1}\beta=\)

Values of \(x\) satisfying \(\sin^{-1}(x^2-5x+7)=2\tan^{-1}1\):

If $(x-a)\cos\theta + y\sin\theta = (x-a)\cos\phi + y\sin\phi = a$, $\tan\frac{\theta}{2} - \tan\frac{\phi}{2} = 2e$ and $\theta, \phi$ are unequal angles less than $360°$, then $y^2$ is equal to:

\(f(x)=\cot^{-1}\!\sqrt{x(x+3)}+\cos^{-1}\!\sqrt{x^2+3x+1}\) is defined on set \(S\). \(S\) equals:

If $E = \cos^2 71° + \cos^2 49° + \cos 71° \cos 49°$, then the value of $10E$ is equal to

All \(x\) satisfying \((\sin^{-1}x)^2-(\cos^{-1}x)^2>0\):

\tan^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \text{ is equal to}

The area bounded by the curve $y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)|$ and the $x$-axis, where $\frac{\pi}{2} \leq x \leq \pi$, is equal to

When the elevation of the sun changes from $45°$ to $30°$, the shadow of a tower increases by 60 units, then the height of the tower is