If $\cot ^{-1}(\alpha)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18+\cot ^{-1} 32+\ldots$ up to $100$ terms,then $\alpha$ is

$4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}$ is equal to

If $x_1, x_2, x_3$ are the roots of the equation $x^3 - x^2 \tan \theta + x \tan^2 \theta + \tan \theta = 0$ and $0 < \theta < \frac{\pi}{4}$, then the value of $\tan^{-1} x_1 + \tan^{-1} x_2 + \tan^{-1} x_3$ at $\theta = \frac{\pi}{12}$ is

If $\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=\sin ^{-1} \alpha$,then $\alpha=$

$2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{5 \sqrt{2}}{7}+2 \tan ^{-1} \frac{1}{8}=$

If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with common difference $d$, then $\tan [\tan^{-1} (\frac{d}{1 + a_1a_2}) + \tan^{-1} (\frac{d}{1 + a_2a_3}) + \dots + \tan^{-1} (\frac{d}{1 + a_{n-1}a_n})] = $