If $\sin x + \sin^2 x = 1,$ then $\cos^8 x + 2\cos^6 x + \cos^4 x = $

In a triangle $ABC$,if $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3 \cos A = 1$,then $\sin (A + B)$ is equal to

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta \}$ be two sets. Then

Let $S = \{\theta \in [0, 2\pi] : 8^{2 \sin^2 \theta} + 8^{2 \cos^2 \theta} = 16\}$. Then $n(S) + \sum_{\theta \in S} \left(\sec \left(\frac{\pi}{4} + 2\theta\right) \operatorname{cosec} \left(\frac{\pi}{4} + 2\theta\right)\right)$ is equal to.

$\sinh ^{-1} 2 + \sinh ^{-1} 3 = x \Rightarrow \cosh x$ is equal to

If $\sin \theta \cosh \alpha = \tan x$ and $\cos \theta \sinh \alpha = \sec x$,then find the value of $\cos 2 \theta \cosh 2 \alpha$.