The value of $\sqrt{2+\sqrt{2(1+\cos 4A)}}$ is equal to

If $\alpha, \beta, \gamma \in \left( 0, \frac{\pi}{2} \right)$,then $\frac{\sin(\alpha + \beta + \gamma)}{\sin \alpha + \sin \beta + \sin \gamma}$ is

If $\cos (\alpha + \beta ) = \frac{4}{5}, \sin (\alpha - \beta ) = \frac{5}{13}$ and $\alpha, \beta$ lie between $0$ and $\frac{\pi}{4}$,then $\tan 2\alpha = $

$\sec {50^o} + \tan {50^o}$ is equal to

The value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ is

If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n,$ then ${a^2} + {b^2} = $