Evaluate $\cot \left(\sum _{n=1}^{19}{\cot }^{-1}\left(1+\sum _{p=1}^{n}2p\right)\right)$ :

Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)=\frac{\pi }{2},$ then the value of $x$ is equal to

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)-\frac{1}{2}{\tan }^{-1}x=0,$ for $x>0,$ then $x=$

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\cot }^{-1}\left(\sqrt{\cos \alpha }\right)-{\tan }^{-1}\left(\sqrt{\cos \alpha }\right)=x,$ then $\sin x$ is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

The number of real roots of the equation ${\tan }^{-1}\sqrt{x\left(x+1\right)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{4}$ is:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If $4{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}x+6{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}x=3\pi ,$ then $x=$

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

Which of the following values of $x$ does not satisfy the equation $\sin \left(2{\cos }^{-1}\left(\cot \left(2{\tan }^{-1}x\right)\right)\right)=0$?

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

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

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\sin }^{-1}x-{\cos }^{-1} x=\frac{\pi }{6}$, then $x$ is

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

The value of $10\tan \left({\cot }^{-1}3+{\cot }^{-1}7\right)$ equal to 

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

The value of $\cos \left({\cos }^{-1}\frac{1}{5}+2{\sin }^{-1}\frac{1}{5}\right)$ is equal to

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If $\cot \left({\sin }^{-1}x\right)=\cos \left({\tan }^{-1}\sqrt{3}\right),$ then $x=$

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

The value of ${\tan }^{-1}\left[\frac{\sqrt{1+x^2}+ \sqrt{1-x^2}}{\sqrt{1+x^2}- \sqrt{1-x^2}}\right],$ $\left|x\right|<\frac{1}{2}, x\ne 0,$ is equal to:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

What is the value of $\text{'}x\text{'},$ if $3{\tan }^{-1}\{1/(2+\sqrt{3}\left)\right\}{\tan }^{-1}x={\tan }^{-1}(1/3)?$

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\cot }^{-1}\left(\alpha \right)={\cot }^{-1}2+{\cot }^{-1}8+{\cot }^{-1}18+{\cot }^{-1}32+\dots .$ upto $100$ terms, then $\alpha$ is:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

Let $f\left(x\right)=\cos \left(2{\tan }^{-1}\sin \left({\cot }^{-1}\sqrt{\frac{1-x}{x}}\right)\right),0<x<1$. Then:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation ${\cos }^{-1}\left(x\right)-2{\sin }^{-1}\left(x\right)={\cos }^{-1}\left(2x\right)$ is equal to

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\cos }^{-1}\left(\frac{5}{13}\right)-{\sin }^{-1}\left(\frac{12}{13}\right)={\cos }^{-1}x$, then $x$ is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\sin }^{-1}x+{\tan }^{-1}x=\pi /2$, then $2x^2+1=$

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

If ${\cos }^{-1}x-{\cos }^{–1}\frac{y}{2}=\alpha ,$ where $-1\le x\le 1, -2\le y\le 2, x\le \frac{y}{2},$ then for all $x, y, 4x^2-4xy\cos \alpha +y^2$ is equal to :

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Inverse Trigonometric Equations and Inequalities

A value of $x$ satisfying the equation $\sin \left[{\cot }^{-1}\left(1+x\right)\right]=\cos \left[{\tan }^{-1}x\right],$ is: