If $A+B+C=\frac{\pi}{4}$,then $\sin 4A+\sin 4B+\sin 4C=$

The expression $\sin x + \sqrt{3} \cos x$ is maximum when $x = \dots \, ^\circ$.

The number of integral values of $k$,for which the equation $7 \cos x + 5 \sin x = 2k + 1$ has a solution,is

The maximum value of the expression $E = \sin \theta + \cos \theta + \sin 2\theta$ is-

If $A + B + C = 180^o$,then $\sum \tan \frac{A}{2} \tan \frac{B}{2} = $

If $ABC$ is not a right-angled triangle and $\sin \left(\frac{\pi}{4}-A\right) \sin \left(\frac{\pi}{4}-B\right) = -\frac{1}{2 \sqrt{2}} \operatorname{cosec}\left(\frac{\pi}{4}-C\right)$,then $\tan A \tan B + \tan B \tan C + \tan C \tan A = $