Trigonometric Series

$96 \cos \frac{\pi }{33} \cos \frac{2\pi }{33} \cos \frac{4\pi }{33} \cos \frac{8\pi }{33} \cos \frac{16\pi }{33}$ is equal to 

Find the value of $96·\cos \frac{\mathrm{\pi }}{33}·\cos \frac{2\mathrm{\pi }}{33}·\cos \frac{4\mathrm{\pi }}{33}.........\cos \frac{16\mathrm{\pi }}{33}$

The sum of the series $\sin \theta +\sin \left(\frac{n-4}{n-2}\right)\theta +\sin \left(\frac{n-6}{n-2}\right)\theta +.....$upto $n$ terms is equal to

If $e^{i\theta }=\mathrm{cis}\theta$ then $\sum _{n=0}^{\infty }\frac{\cos \left(n\theta \right)}{2^n}=$

If $\sin \theta +\sin 3\theta +\sin 5\theta +...+\sin \left(2n-1\right)\theta =\lambda \frac{{\sin }^{2}n\theta }{\sin \theta }$, then value of $\lambda$ is

The value of $\cos \frac{\pi }{2^2}\cos \frac{\pi }{2^3}\mathrm{.....}\cos \frac{\pi }{2^{10}}\sin \frac{\pi }{2^{10}}$ is

$\sin \left(\frac{19\mathrm{\pi }}{34}\right)\sin \left(\frac{21\mathrm{\pi }}{34}\right)\sin \left(\frac{9\mathrm{\pi }}{34}\right)\sin \left(\frac{33\mathrm{\pi }}{34}\right)=$

If the value of $\cos \frac{2\pi }{15}·\cos \frac{4\pi }{15}·\cos \frac{8\pi }{15}·\cos \frac{14\pi }{15}$ is $\frac{1}{k}$ then find $k.$

Let $p_{\mathit{n}}\mathit{=}cos\left(\frac{{\mathit{2}}^{\mathit{n}}\mathit{\pi }}{\mathit{65}}\right)$ and ${\mathrm{q}}_{\mathrm{n}}=\cos \left(\frac{(4\mathrm{n}-2)\pi }{65}\right),$ where $\mathrm{n}\in \mathrm{N}.$ If $\mathrm{S}=\left(\frac{{\sum }_{\mathrm{n}=1}^{16}{\mathrm{q}}_{\mathrm{n}}}{{\prod }_{\mathrm{n}=1}^5{\mathrm{p}}_{\mathrm{n}}}\right)$ then find the value of $S.$

If ${\cos }^2\theta +{\cos }^3\left(\frac{2\pi }{3}+\theta \right)+{\cos }^3\left(\frac{4\pi }{3}+\theta \right)=a\cos 3\theta ,$ then $a=$

Given, $\cos 2^m\theta \cos 2^{m+1}\theta \dots ..\cos 2^n\theta =\frac{\sin 2^{n+1} \theta }{2^{n- m+1}\sin 2^m\theta }$ , where $2^m\theta \ne k\pi , n, m\in I,$ then $\left(\cos 2^3\frac{\pi }{10}\cos 2^4\frac{\pi }{10}\cos 2^5\frac{\pi }{10}\dots ..\cos 2^{10}\frac{\pi }{10}\right)$ is equal to

The value of $\left(1+\sec 2\theta \right) \left(1+\sec 2^2\theta \right) \left(1+\sec 2^3\theta \right)\dots \left(1+\sec 2^n\theta \right)$ (whenever defined) is equal to

The value of $\sum _{r=1}^{18}{\cos }^2\left(5r\right)°$, where $x°$ denotes the $x$ degrees, is equal to

$\cos 40°\cos 80°\cos 160°=$

The value of $\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}\dots .$ upto $n$ terms is equal to

The value of $\cos 20°+\cos 40°+......+\cos 140°$ is not equal to

The value of $\cos 20° + \cos 40° +......+ \cos 140°$ is not equal to

The value of$\mathrm{ }\cos \frac{\mathrm{\pi }}{65}\cos \frac{2\mathrm{\pi }}{65}\mathrm{c}\mathrm{o}\mathrm{s}\frac{4\mathrm{\pi }}{65}\dots \cos \frac{32\mathrm{\pi }}{65}$ is