In a $\triangle ABC$,suppose none of the angles are multiples of $\frac{\pi}{2}$,then what is the value of $\cot A \cot B + \cot B \cot C + \cot A \cot C$?

If $\sin \theta + \sqrt{3} \cos \theta = 6x - x^2 - 11$,where $x \in R$ and $0 \le \theta \le 2\pi$,then the equation has a solution for:

If $A + B + C = \pi$ and $\sin \left( A + \frac{C}{2} \right) = k \sin \frac{C}{2}$,then $\tan \frac{A}{2} \tan \frac{B}{2} = $

The maximum value of the function $f(x) = 3 \sin^{12} x + 4 \cos^{16} x$ is

Assertion $(A)$: If $A=10^{\circ}, B=16^{\circ}, C=19^{\circ}$,then $\tan 2A \tan 2B + \tan 2B \tan 2C + \tan 2C \tan 2A = 1$.
Reason $(R)$: If $A+B+C=90^{\circ}$,then $\tan A \tan B + \tan B \tan C + \tan C \tan A = 1$.
Which of the following is correct?

If $A + B + C = \pi ,$ then $\tan ^2 \frac{A}{2} + \tan ^2 \frac{B}{2} + \tan ^2 \frac{C}{2}$ is always