$\sinh (x+y) \cosh (x-y)$ is equal to

If $\tan \left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{x}{2}\right)$,then $\frac{3 \sin x+\sin ^3 x}{1+3 \sin ^2 x}=$

If $\sin x + \text{cosec } x = 2$,then $\sin^n x + \text{cosec}^n x$ is equal to

$\cos ^2 10^{\circ}+\cos ^2 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}$ is equal to

If $\tan \left(\frac{\pi}{4}+\alpha\right)=\tan ^3\left(\frac{\pi}{4}+\beta\right)$,then $\tan (\alpha+\beta) \cot (\alpha-\beta)=$

Evaluate the product: $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{2 \pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{4 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{6 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$