Let $0 < \alpha < 1$, $\beta = \frac{1}{3\alpha}$, and $\tan^{-1}(1 - \alpha) + \tan^{-1}(1 - \beta) = \frac{\pi}{4}$. Then $6(\alpha + \beta)$ is equal to:

If ${\cot ^{ - 1}}\frac{n}{\pi } > \frac{\pi }{6},\,\,n \in N$,then the maximum value of $n$ is

If $\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}$,then $\cos^{-1} x + \cos^{-1} y = $

The value of $\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]$ is

The value of $\cot \left(\sum_{n=1}^{23} \cot ^{-1}\left(1+\sum_{k=1}^n 2 k\right)\right)$ is

Find the sum of the series: $\tan^{-1} \left( \frac{1}{1 + 1 \times 2} \right) + \tan^{-1} \left( \frac{1}{1 + 2 \times 3} \right) + \dots + \tan^{-1} \left( \frac{1}{1 + n(n + 1)} \right) =$