If $\pi < \alpha < \frac{3\pi}{2}$,then $\sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} + \sqrt{\frac{1 + \cos \alpha}{1 - \cos \alpha}} = $

If $0 < \theta < \frac{\pi}{2}$ and $\tan 3 \theta \neq 0$,then $\tan \theta + \tan 2 \theta + \tan 3 \theta = 0$ if $\tan \theta \cdot \tan 2 \theta = k$,where $k =$

If $2 \sin \theta + 3 \cos \theta = 2$ and $\theta \neq (2n + 1) \frac{\pi}{2}$,then find the value of $3 \sin \theta - 2 \cos \theta$.

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 $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $A$.$P$.,then

If $\sin \theta + \cos \theta = m$ and $\sec \theta + \text{cosec} \theta = n$,then $n(m + 1)(m - 1) = $