CUET Mathematics Geometry > Trigonometry PYQs

The value of $\tan^{-1}(2) + \tan^{-1}(3)$ is equal to

The value of $-\cosec^2(\cot^{-1}y) + \sec^2( \tan^{-1}x)$ is equal to

The simplified form of $\tan^{-1}\left(\frac{\cos x}{1+\sin x}\right)$, $-\frac{\pi}{2} < x < \frac{\pi}{2}$ is

$\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ equals:

Consider $f(x) = \sin(3x) + 4, \forall x \in \mathbb{R}$. Then (A) Maximum value of $f(x)$ is 5 (B) Minimum value of $f(x)$ is 3 (C) Maximum value of $f(x)$ is attained at $x = \frac{\pi}{6}$ (D) Minimum value of $f(x)$ is attained at $x = 0$ Choose the correct answer from the options given below:

The value of $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24}$ | (I) $\frac{3\pi}{4}$ | | (B) $\tan^{-1}2 + \tan^{-1}3$ | (II) $\pi$ | | (C) $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3$ | (III) $\tan^{-1}\frac{1}{2}$ | | (D) $\tan^{-1}\frac{1}{7} + \tan^{-1}\frac{1}{13}$ | (IV) $\tan^{-1}\frac{2}{9}$ | Choose the correct answer from the options given below:

Arrange the principal values of the following functions in ascending order (A) $\cosec^{-1}(2)$ (B) $\tan^{-1}(-\sqrt{3})$ (C) $\tan^{-1}(1)$ (D) $\tan^{-1}\left(\cos\frac{3\pi}{7}\right)$ Choose the correct answer from the options given below:

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is

The value of $\tan^2(\sec^{-1} 2) + \cot^2(\cosec^{-1} 3)$ is equal to

The minimum value of the function $f(x) = 3\sin x - 4\cos x, x \in [-4\pi, 4\pi]$ is equal to

$\sin^{-1}(\cos\frac{3\pi}{5})$ equals

Match List-I with List-II | List-I | List-II | |---|---| | (Inverse Trigonometric Function) | (Principal Value) | | (A) $\sin^{-1}(-\frac{1}{2})$ | (I) ${\pi}/{6}$ | | (B) $\cos^{-1}(-\frac{1}{2})$ | (II) $-{\pi}/{6}$ | | (C) $\tan^{-1}(-\sqrt{3})$ | (III) ${2\pi}/{3}$ | | (D) $\cot^{-1}(\sqrt{3})$ | (IV) $-{\pi}/{3}$ | Choose the correct answer from the options given below:

The maximum value of $\sin x \cdot \cos x$ is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\cos^{-1} x + \cos^{-1}(-x)$ | (I) $\frac{\pi}{3}$ | | (B) $\text{cosec}^{-1}(-x) + \sec^{-1}(-x)$ | (II) $-\frac{\pi}{3}$ | | (C) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (III) $\pi$ | | (D) $\tan^{-1}\left(\tan\frac{4\pi}{3}\right)$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Inverse Trigonometric function | Principal values of arguments | | (A) $sin^{-1}\left(\frac{-1}{2}\right)$ | (I) $\frac{-\pi}{3}$ | | (B) $cos^{-1}\left(\frac{-1}{2}\right)$ | (II) $\frac{3\pi}{4}$ | | (C) $tan^{-1}(-\sqrt{3})$ | (III) $\frac{-\pi}{6}$ | | (D) $sec^{-1}(-\sqrt{2})$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

For the principal value branch, the value of $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$ is

For $x \in [-1,1]$, if $4\sin^{-1}x + \cos^{-1}x = \pi$ then $x$ is equal to

for $|x| < 1$, $sin (tan^{-1}x)$ equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\sin^{-1}(-1)$ | (I) $\frac{5\pi}{6}$ | | (B) $\cot^{-1}(-1)$ | (II) $\frac{-\pi}{2}$ | | (C) $\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ | (III) $\frac{\pi}{4}$ | | (D) $\tan^{-1}(1)$ | (IV) $\frac{3\pi}{4}$ | Choose the correct answer from the options given below:

The value of $\tan^{-1}(1) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ is

If $\tan ^{-1}\left(\frac{2}{3^{-x}+1}\right)=\cot ^{-1}\left(\frac{3}{3^{x}+1}\right)$, then which one of the following is true ?

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$.

The maximum value of $\sin x + \cos x, x \in R$ is:

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

The principal value of $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is :

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

The maximum value of $(\sin x)(\cos x)$ is :

Let $a \leq \tan^{-1} x + \cot^{-1} x + \sin^{-1} x \leq b$. If $\alpha$ and $\beta$ denote the minimum and maximum possible values of a and b respectively, then :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) Range of $y = \text{cosec}^{-1}x$ | (I) $R - (-1, 1)$ | | (B) Domain of $\sec^{-1}x$ | (II) $(0, \pi)$ | | (C) Domain of $\sin^{-1}x$ | (III) $[-1, 1]$ | | (D) Range of $y = \cot^{-1}x$ | (IV) $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | Choose the correct answer from the options given below :

Let $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Then $y$ is :

The value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (I) $\frac{3\pi}{4}$ | | (B) $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ | (II) $-\frac{\pi}{3}$ | | (C) $\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)$ | (III) $\frac{\pi}{2}$ | | (D) $\cos^{-1}\left(\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below :

The value of $\cos^{-1} \left( \sin \left( \cos^{-1} \frac{1}{2} \right) \right) + \tan^{-1}(1)$

The Principal value of $\cos^{-1} \left( -\frac{1}{2} \right)$ is:

Match List - I with List - II | List - I | List - II | |---|---| | A. $\tan^{-1}\left(\tan\frac{7\pi}{6}\right)$ | I. $\frac{5\pi}{6}$ | | B. $\tan^{-1}\left(\tan\frac{8\pi}{6}\right)$ | II. $\frac{\pi}{2}$ | | C. $\tan^{-1}\frac{1}{\sqrt{3}} + cosec^{-1}\frac{2}{\sqrt{3}}$ | III. $\frac{\pi}{6}$ | | D. $\cos^{-1}\left(\cos\frac{5\pi}{6}\right)$ | IV. $\frac{\pi}{3}$ | Choose the correct answer from the options given below:

The value of $\sec^2(\tan^{-1}2) + cosec^2(\cot^{-1}3)$ is:

$\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}$, than $x$ is equal to (a) $0$ (b) $1$ (c) $\frac{1}{2}$ (d) $2$ Choose the most appropriate answer from the options given below:

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

Match List I with List II | List I (Functions) | List II (Principal value branches) | |---|---| | A. $f(x) = \cos^{-1} x$ | I. $[0, \pi]$ | | B. $f(x) = \tan^{-1} x$ | II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$ | | C. $f(x) = \text{cosec}^{-1} x$ | III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | | D. $f(x) = \sec^{-1} x$ | IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | Choose the correct answer from the options given below:

The graph of $\sin^{-1} x$ is represented by:

The value of $\text{cosec}^{-1}(-2) - 2\sec^{-1}(-2)$ is equal to:

$\sin^{-1}(\cos x) = \frac{\pi}{2} - x$ is valid for :

Match List I with List II | LIST I | LIST II | |---|---| | A. $\sin^{-1} x + \cos^{-1} x, x \in [-1,1]$ | I. $-\frac{\pi}{2}$ | | B. $\tan^{-1} \sqrt{3} - \cot^{-1}(-\sqrt{3})$ | II. $-\frac{\pi}{6}$ | | C. $\cos^{-1}\left(\cos\frac{13\pi}{6}\right)$ | III. $\frac{\pi}{2}$ | | D. $\sin^{-1}\left(-\frac{1}{2}\right)$ | IV. $\frac{\pi}{6}$ | Choose the correct answer from the options given below:

Value of $\frac{e^{\sin(\tan^{-1} x + \cot^{-1} x)}}{e^{\sin(\sin^{-1} x + \cos^{-1} x)}}, x \in [-1, 1]$, is: