$2 \sin^2 \beta + 4 \cos(\alpha + \beta) \sin \alpha \sin \beta + \cos 2(\alpha + \beta) = $

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

Evaluate the expression: $\sum \frac{1}{1 + x^{a-b} + x^{a-c}}$

Assertion $(A)$: If $\sqrt{4 \sin^4 \theta + \sin^2 2\theta} + 4 \cos^2\left(\frac{\pi}{4} - \frac{\theta}{2}\right) = 2$,then $\theta$ lies in the $3^{\text{rd}}$ quadrant or $4^{\text{th}}$ quadrant.
Reason $(R)$: $\sqrt{\sin^2 \theta} = \sin \theta$

If $\alpha_1, \alpha_2, \cdots, \alpha_n$ are in $A$.$P$. with common difference $\theta$, then the sum of the series $\sec \alpha_1 \sec \alpha_2 + \sec \alpha_2 \sec \alpha_3 + \cdots + \sec \alpha_{n-1} \sec \alpha_n = k(\tan \alpha_n - \tan \alpha_1)$, where $k=$

$\tan \frac{\pi}{5} + 2 \tan \frac{2 \pi}{5} + 4 \cot \frac{4 \pi}{5} = $