If $\cos (A + B) = \alpha \cos A \cos B + \beta \sin A \sin B,$ then $(\alpha, \beta) =$

If $\tan \alpha = (1 + 2^{-x})^{-1}$ and $\tan \beta = (1 + 2^{x+1})^{-1}$,then $\alpha + \beta$ equals:

$\cos \left(\frac{3 \pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right) = $

If $\tan A = \frac{5}{6}$ and $\tan B = \frac{1}{11}$,then $A + B = $

The value of $\tan \frac{2\pi}{5} - \tan \frac{\pi}{15} - \sqrt{3} \tan \frac{2\pi}{5} \tan \frac{\pi}{15}$ is equal to

If two angles $\alpha, \beta$ are such that $0 < \alpha, \beta < \frac{\pi}{4}$,$\sqrt{1+\cos 2 \alpha}=\frac{3}{\sqrt{5}}$ and $\frac{\sqrt{1-\cos 2 \beta}}{\sqrt{1+\cos 2 \beta}}=\frac{1}{7}$,then $(2 \alpha+\beta)=$