If $\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2a$,then the value of $\left(a+\frac{1}{a}\right)$ is:

If $0 \le x \le \pi$ and $81^{\sin^2 x} + 81^{\cos^2 x} = 30$,then $x =$

If $x=\log _e\left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$ and $\theta \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$,then consider the following statements:
$I$ : $\cosh x=\sec 2 \theta$
$II$ : $\sinh x=-\tan 2 \theta$
Then which one of the following options is true?

If $\cos A = m \cos B,$ then

$\tan 81^{\circ}-\tan 63^{\circ}-\tan 27^{\circ}+\tan 9^{\circ}$ is equal to

If $\cos x + \sec x = -2$,then for a positive integer $n$,$\cos^n x + \sec^n x$ is