If $\theta = \frac{\pi}{9}$,then $1 + 27 \tan^2 \theta - 33 \tan^4 \theta + \tan^6 \theta = $

$\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdots \cos \frac{\pi}{2^{10}} = $

If $\sin 6 \theta = 32 \cos^5 \theta \sin \theta - 32 \cos^3 \theta \sin \theta + 3x$,then $x$ is equal to:

If $\cos ^3 x \sin 4 x = \sum_{r=0}^{n} a_{r} \sin rx$ for all $x \in R$,then $a_3+a_5 : a_1+a_7 = $

If $\tan \alpha = \frac{-12}{5}$,$\cot \beta = \frac{7}{24}$,$\alpha$ does not belong to the second quadrant,and $\beta$ does not belong to the first quadrant,then $\sqrt{13} \sin \frac{\alpha}{2} + \cos \frac{\beta}{2} + \tan \frac{\alpha}{2} \cot \frac{\beta}{2} = $

If $\sec x + \tan x = 2$,where $0 < x < \frac{\pi}{2}$,then find the value of $\sin \frac{x}{4}$.