Question

Find sum of series $cosec\theta +cosec2\theta +cosec4\theta +....$ to $n$ terms

Answer 1

$\csc \theta =\frac{1}{\sin \theta }$

$=\frac{\sin (\theta -\frac{\theta }{2})}{\sin \theta \cdot \sin \frac{\theta }{2}}$

$=\frac{\sin \theta \cos \frac{\theta }{2}-\cos \theta \sin \frac{\theta }{2}}{\sin \theta \cdot \sin \frac{\theta }{2}}$

or $\csc \theta =\cot \frac{\theta }{2}-\cot \theta$

Similarly, $\csc 2\theta =\cot \theta -\cot 2\theta$

$\csc 4\theta =\cot 2\theta -\cot 4\theta$

---------------------------------------------------------

----------------------------------------------------------

$\csc 2^{n-1}\theta =\cot 2^{n-2}\theta -\cot 2^{n-1}\theta$

Adding we get, $\csc \theta +\csc 2\theta +\csc 4\theta +----------$ up to $n$ term$=\cot \frac{\theta }{2}-\cot 2^{n-1}\theta$