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

In an isosceles right-angled triangle,a straight line is drawn from the midpoint of one of the equal sides to the opposite vertex. Then a pair of possible values of the cotangents of the two angles so formed at that vertex are

Prove that: $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13} + \cos \frac{3 \pi}{13} + \cos \frac{5 \pi}{13} = 0$

The value of $\cos 105^\circ + \sin 105^\circ$ is

Prove that $\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \cot x$.

If $\tan A = \frac{1}{2}$ and $\tan B = \frac{1}{3}$,then $\cos 2A = $