The variable $x$ satisfying the equation $|\sin x \cos x| + \sqrt{2 + \tan^2 x + \cot^2 x} = \sqrt{3}$ belongs to the interval

If $\cos(x) + \cos^2(x) = 1$,then $\sin^2(x) + \sin^4(x)$ is equal to

If $\sin x + \cos x = a$,where $a \in [-\sqrt{2}, \sqrt{2}] - \{-1, 1\}$,then $\sum_{n=1}^{\infty} (\sin^n x + \cos^n x)$ is equal to -

Let $\frac{\pi}{2} < \theta < \pi$ and $\cot \theta = -\frac{1}{2 \sqrt{2}}$. Then the value of $\sin (\frac{15 \theta}{2}) (\cos 8 \theta + \sin 8 \theta) + \cos (\frac{15 \theta}{2}) (\cos 8 \theta - \sin 8 \theta)$ is equal to:

If $\theta$ is an acute angle and $2 \sin ^2 \theta = \cos ^4 \frac{\pi}{8} + \sin ^4 \frac{3 \pi}{8} + \cos ^4 \frac{5 \pi}{8} + \sin ^4 \frac{7 \pi}{8}$,then $\theta =$

$\cos 13^{\circ} \sin 17^{\circ} \sin 21^{\circ} \cos 47^{\circ} = $