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

If $\tan \theta = \frac{\sin \alpha - \cos \alpha}{\sin \alpha + \cos \alpha}$,then $\sin \alpha + \cos \alpha$ and $\sin \alpha - \cos \alpha$ must be equal to

If $2\tan A = 3\tan B,$ then $\frac{\sin 2B}{5 - \cos 2B}$ is equal to

If $\sin \theta + \cos \theta = \sqrt{2} \cos \theta$,then the value of $(\cos \theta - \sin \theta)$ is

$A$ ladder is placed along a wall such that its upper end is touching the top of the wall. The foot of the ladder is $10 \, ft$ away from the wall and the ladder makes an angle of $60^{\circ}$ with the ground. When a man starts climbing on it,it slips and now the ladder makes an angle of $30^{\circ}$ with the ground. How much did the ladder slip down from the top of the wall? (in $ft$)

The value of $\cos^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{4} + \cos^2 \frac{5\pi}{12}$ is