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 $\sin^3 x \sin 3x = \sum_{m=0}^n c_m \cos mx$ where $c_0, c_1, c_2, \dots, c_n$ are constants and $c_n \neq 0$,then the value of $n$ is

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

If $90^\circ < A < 180^\circ$ and $\sin A = \frac{4}{5}$,then $\tan \frac{A}{2}$ is equal to

If $\frac{\sec 8\theta - 1}{\sec 4\theta - 1} = \frac{a + b \tan^2 2\theta}{1 + c \tan^2 2\theta + d \tan^4 2\theta}$ (where $\theta \neq \frac{n\pi}{16}, n \in I$),then the value of $(a - b + c - d)$ is -

If the identity $\cos^4 \theta = a \cos 4\theta + b \cos 2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$,then $(a, b, c) =$