If $0<\mathrm{a}, \mathrm{b}<1$, and $\tan ^{-1} \mathrm{a}+\tan ^{-1} \mathrm{~b}=\frac{\pi}{4}$, then the value of $(a+b)-\left(\frac{a^2+b^2}{2}\right)+\left(\frac{a^3+b^3}{3}\right)-\left(\frac{a^4-b^4}{4}\right)+\ldots$ is : (1) $\log _e 2$ (2) $\mathrm{e}^2-1$

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

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

The number of solutions of the equation $$ \sin ^{-1}\left[x^2+\frac{1}{3}\right]+\cos ^{-1}\left[x^2-\frac{2}{3}\right]=x^2 $$ for $x \in[-1,1]$ and $[x]$ denotes the greatest integer less than or equal...

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy $$ \sin ^{-1}\left(\frac{3 x}{5}\right)+\sin ^{-1}\left(\frac{4...

The value of $\left(2 \tan ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)\right)$ is equal to:

A possible value of $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is :

The value of $\cot \left(\sum_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is :

The value of $\lim _{n \rightarrow \infty} 6 \tan \left\{\sum_{r=1}^n \tan ^{-1}\left(\frac{1}{r^2+3 r+3}\right)\right\}$