Find the value of $\cos \left({\cos }^{-1}\left(\frac{4}{5}\right)+{\sin }^{-1}\left(\frac{4}{5}\right)\right)$.

Important Questions on Inverse Trigonometric Functions

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

${\cot }^{-1}\left(21\right)+{\cot }^{-1}\left(13\right)+{\cot }^{-1}(-8)=$

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

$\cos \left[2{\sin }^{-1}\frac{3}{4}+{\cos }^{-1}\frac{3}{4}\right]=$

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If ${\sin }^{-1}x+{\cos }^{-1}y=2\pi /5$, then ${\cos }^{-1}x+{\sin }^{-1}y$ is

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

$cosec\left[2{\cot }^{-1}\left(5\right)+{\cos }^{-1}\left(\frac{4}{5}\right)\right]$ is equal to:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If $x={\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left(\sin 10\right)\mathrm{ }$ and $y={\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\cos 10\right),$ then $y-x$ is equal to:

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If $\frac{{\sin }^{-1}x}{a}=\frac{{\cos }^{-1}x}{b}=\frac{{\tan }^{-1}y}{c}; 0<x<1,$ then the value of $\cos \left(\frac{\pi c}{a+b}\right)$ is:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

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

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

${\cot }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)=$

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The solution of ${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$ is

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If $f^{\text{'}}\left(x\right)={\tan }^{-1}\left(\sec x+\tan x\right),-\frac{\pi }{2}<x<\frac{\pi }{2}$ and $f\left(0\right)=0$ , then $f\left(1\right)$ is equal to:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

$\cos \left(2\left({\tan }^{-1}\frac{1}{5}+{\tan }^{-1}5\right)\right)=\_\_\_\_\_\_\_.$

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

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

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

${\tan }^{-1}\left(\cot x\right)+{\cot }^{-1}\left(\tan x\right)=$ (where, $0<x<\frac{\pi }{2}$)

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Assume that $3.13\le \pi \le 3.15.$ The integer closest to the value of ${\sin }^{-1}(\sin 1\cos 4+\cos 1 \sin 4),$ where $1$ and $4$ appearing in $\sin$ and $\cos$ are given in radians, is:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

${\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)+{\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)=$

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If ${\sin }^{-1}x{\cos }^{-1}x=p$, then the value of ${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3$ is

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of ${\cos }^{-1}\left(\cos \frac{5\pi }{4}\right)$ is

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

If $a+\frac{\pi }{2}<2 {\tan }^{-1}x+3 {\cot }^{-1}x<b$ then $\text{'}a\text{'}$ and $\text{'}b\text{'}$ are respectively.

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The principal value of ${\tan }^{-1}\left(\cot \frac{43\pi }{4}\right)$ is

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}=$