The expression $\cos^2 \phi + \cos^2(\theta + \phi) - 2 \cos \theta \cos \phi \cos(\theta + \phi)$ is

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 $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=$

Evaluate: $\sin \frac{\pi}{12} \sin \frac{2 \pi}{12} \sin \frac{3 \pi}{12} \sin \frac{4 \pi}{12} \sin \frac{5 \pi}{12} \sin \frac{6 \pi}{12}$

If $|\cos x + \sin x| + |\cos x - \sin x| = 2 \sin x$ for $x \in [0, 2\pi]$,then the maximum integral value of $x$ is:

If the orthocentre and circumcentre of a triangle $ABC$ are at equal distances from the side $BC$ and lie on the same side of $BC$,then the value of $\tan B \tan C$ is equal to: