Basics of Inverse Trigonometric Functions

Let $f\left(x\right)={\sec }^{-1}\left(x-10\right)+{\cos }^{-1}\left(10-x\right)$. The range of $f\left(x\right)$ is

The domain and range of $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x+{\cot }^{-1}x+{\sec }^{-1}x+{cosec}^{-1}x$  respectively are -

If $k$ satisfies $k^2-k-6>0$, then a value exists for -

If ${\cos }^{-1}a+{\cos }^{-1}b+{\cos }^{-1}c=\pi$, then the value of $\left(a\sqrt{1-a^2}+b\sqrt{1-b^2}+c\sqrt{1-c^2}\right)$ will be

The solution of the inequality ${\left({\tan }^{-1}x\right)}^2-3{\tan }^{-1}x+2\ge 0$ is -

${\left({\sin }^{-1}x\right)}^2+{\left({\sin }^{-1}y\right)}^2+2\left({\sin }^{-1}x\right)\left({\sin }^{-1}y\right)={\pi }^2$, then $x^2+y^2$ is equal to -

If $a=\frac{1}{4}+i\frac{\sqrt{3}}{4}$ and $z=x+iy,$ then ${\sin }^{-1}|z{|}^2+{\cos }^{-1}(a\overline{z}+\overline{a}z-2)$ equals to

${\sin }^{-1}\left(\frac{x^2}{4}+\frac{y^2}{9}\right)+{\cos }^{-1}\left(\frac{x}{2\sqrt{2}}+\frac{y}{3\sqrt{2}}-2\right)$ equals to

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

If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\pi }{2},$ then the value of  $x^{100}+y^{100}+z^{100}-\frac{9}{x^{101}+y^{101}+z^{101}}$ is equal to

For the equation ${\cos }^{-1}x+{\cos }^{-1}2x+\pi =0$, the number of real solutions is

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

The trigonometric equation ${\sin }^{-1}x=2 {\sin }^{-1}2a$ has a real solution if

Which of the following is the domain of the function $f\left(x\right)=2\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}\sqrt{1-x}+\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}\left(2\sqrt{x\left(1-x\right)}\right)$?

The sum of the solutions of the equation $2{\sin }^{-1}\sqrt{x^2+x+1}+{\cos }^{-1}\sqrt{x^2+x}=\frac{3\pi }{2}$ is

Statement $\mathrm{I}$: $\underset{x\to 0}{\lim }\left[\frac{{\tan }^{-1}x}{x}\right]=0$, where, $\left[·\right]$ represents the greatest integer function.

Statement $\mathrm{II}$: $\left(\frac{{\tan }^{-1}x}{x}\right)<1$ in the neighbourhood of $x=0$.

Find the value of $xy+yz+zx$, if  ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=3\pi .$

The number of integral values in the range of the function $f\left(x\right)={\sin }^{-1}x-{\cot }^{-1}x+x^2+2x+6$ is

The complete set of values of $a$ for which the function $f\left(x\right)={\tan }^{-1}\left(x^2-18x+a\right)>0; \forall x\in R$

$\tan \left(\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}\alpha \right)+\tan \left(\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}\alpha \right), x\ne 0,$ is equal to :