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

If $\cos x + \cos y = -\cos \alpha$ and $\sin x + \sin y = -\sin \alpha$,then $\cot \left(\frac{x+y}{2}\right) = $

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

If $\sin \theta + \cos \theta = p$ and $\tan \theta + \cot \theta = q$,then $q(p^2 - 1)$ is equal to

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$ for all $x \in R$,then $f(2023) = $

Which of the following statements is/are correct for $0 < \theta < \frac{\pi}{2}$?