The maximum and minimum values of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 5 \cos x + 3 \cos \left(x + \frac{\pi}{3}\right) + 8$ for all $x \in \mathbb{R}$,are respectively.

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?

Let $x, y, z$ be real numbers such that $x \geq y \geq z \geq \frac{\pi}{12}$. If $x+y+z = \frac{\pi}{2}$,then the minimum value of $\cos x \cdot \sin y \cdot \cos z$ is

If $5 \cos x + 12 \cos y = 13$,then the maximum value of $5 \sin x + 12 \sin y$ is:

The maximum value of $a \cos x + b \sin x$ is

If $A+B+C=60^{\circ}$,then $\cos (30^{\circ}-A)+\cos (30^{\circ}-B)+\cos (30^{\circ}-C)+\sin (A+B+C) = $