If $\sin x \cosh y = \cos \theta$ and $\cos x \sinh y = \sin \theta$,then $\sin^2 x + \cosh^2 y =$

If $a \sin^2 \theta + b \cos^2 \theta = c$,then $\tan^2 \theta$ is equal to

Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos \beta - \cos \alpha) + \cos \alpha \cos \beta = 1$. Then which of the following is/are true?

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\tan (\alpha+2 \beta)$ is equal to

$\sin \frac{2 \pi}{5}+\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}+\sin \frac{8 \pi}{5}$ is equal to

If $15 \sin^{4} \alpha + 10 \cos^{4} \alpha = 6$ for some $\alpha \in R$,then the value of $27 \sec^{6} \alpha + 8 \operatorname{cosec}^{6} \alpha$ is equal to ....... .