If $\cosh 2x = 241$,then $\operatorname{coth} x$ is equal to

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to

If $\tan (\pi \cos \theta) = \cot (\pi \sin \theta)$,then $\sin \left(\frac{\pi}{4} + \theta\right) =$

Prove that $\cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1$.

Let $\frac{\pi}{2} < \theta < \pi$ and $\cot \theta = -\frac{1}{2 \sqrt{2}}$. Then the value of $\sin (\frac{15 \theta}{2}) (\cos 8 \theta + \sin 8 \theta) + \cos (\frac{15 \theta}{2}) (\cos 8 \theta - \sin 8 \theta)$ is equal to:

In a right-angled triangle,the hypotenuse is $2 \sqrt{2}$ times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are