Let $|\cos \theta \cos (60^{\circ}-\theta) \cos (60^{\circ}+\theta)| \leq \frac{1}{8}$, where $\theta \in [0, 2\pi]$. Then, the sum of all $\theta \in [0, 2\pi]$ where $\cos 3\theta$ attains its maximum value is: (in $\pi$)

If $\cos A+\cos (A+B)+\cos (A+2 B)+\ldots$ up to $n$ terms $=$ $\cos \left(\frac{2 A+(n-1) B}{2}\right) \sin \frac{n B}{2} \operatorname{cosec} \frac{B}{2}$,then $\cos \frac{\pi}{19}+\cos \frac{3 \pi}{19}+\cos \frac{5 \pi}{19}+\ldots+\cos \frac{17 \pi}{19} = $

If the equation $\tan^4x - 2\sec^2x + [a]^2 = 0$ has at least one solution,then the complete range of $a$ (where $a \in R$) is:
(Note: $[k]$ denotes the greatest integer less than or equal to $k$)

If $\tan \alpha = 2 \sin \beta \sin \gamma \operatorname{cosec}(\beta + \gamma)$,then

Let $f(\theta) = \sin \theta (\sin \theta + \sin 3\theta)$,then $f(\theta)$

If $\sin \theta + \cos \theta = p$ and $\sin^3 \theta + \cos^3 \theta = q$,then $p(p^2 - 3)$ is equal to