If $\sinh u = \tan \theta$,then $\cosh u$ is equal to

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

Let $S = \{x \in R : \cos(x) + \cos(\sqrt{2}x) < 2\}$,then

For any positive integer $n$,let $S_n: (0, \infty) \rightarrow R$ be defined by $S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$,where for any $x \in R$,$\cot^{-1} x \in (0, \pi)$ and $\tan^{-1} x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following statements is (are) $TRUE$?
$(A)$ $S_{10}(x) = \frac{\pi}{2} - \tan^{-1}\left(\frac{1+11x^2}{10x}\right)$,for all $x > 0$
$(B)$ $\lim_{n \rightarrow \infty} \cot(S_n(x)) = x$,for all $x > 0$
$(C)$ The equation $S_3(x) = \frac{\pi}{4}$ has a root in $(0, \infty)$
$(D)$ $\tan(S_n(x)) \leq \frac{1}{2}$,for all $n \geq 1$ and $x > 0$

If $a$ and $b$ respectively represent the lengths of a side and a diagonal of a regular pentagon that is inscribed in a circle,then $\frac{b}{a}=$

If $\cosh x = \operatorname{cosec} \theta$,then $\coth^2 \frac{x}{2} = $