The value of $(1-\cos \theta)(1+\cos \theta)(1+\cot^2 \theta)$,when $\theta = \frac{\pi}{15}$ is

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $............$.

Let $a$ and $b$ be non-negative real numbers. If $\sin x + a \cos x = b$,then $|a \sin x - \cos x| = $

Evaluate the sum: $\sin^2(3^{\circ}) + \sin^2(6^{\circ}) + \sin^2(9^{\circ}) + \dots + \sin^2(84^{\circ}) + \sin^2(87^{\circ}) + \sin^2(90^{\circ})$

If $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 46^{\circ} \sin 47^{\circ}}+\ldots$ up to $45$ terms $=\frac{1}{\sin x^{\circ}}$,then $\sin \left(\frac{\pi}{2} x\right)=$

Statement $(S1): \sin 55^{\circ} + \sin 53^{\circ} - \sin 19^{\circ} - \sin 17^{\circ} = \cos 2^{\circ}$.
Statement $(S2):$ The range of $\frac{1}{3 - \cos 2x}$ is $\left[\frac{1}{4}, \frac{1}{2}\right]$.
Which one of the following is correct?