Question

If $cosec\frac{\pi }{32}+cosec\frac{\pi }{16}+cosec\frac{\pi }{8}+cosec\frac{\pi }{4}+cosec\frac{\pi }{2}=cot\frac{\pi }{k}$. Find k.

Answer 1

$\csc \frac{\pi }{32}+\csc \frac{\pi }{16}+\csc \frac{\pi }{8}+\csc \frac{\pi }{4}+\csc \frac{\pi }{2}=\cot \frac{\pi }{K}$

$\Rightarrow \csc \theta +\cot \theta =\cot \frac{\theta }{2}$

$\Rightarrow$ using this property

$LHS$

$\Rightarrow \csc \frac{\pi }{3}+\cos \frac{\pi }{16}+\csc \frac{\pi }{8}+\csc \frac{\pi }{4}+\csc \frac{\pi }{2}+\cot \frac{\pi }{2}-\cot \frac{\pi }{2}$

$\Rightarrow \csc \frac{\pi }{32}+\csc \frac{\pi }{16}+\csc \frac{\pi }{8}+\cot \frac{\pi }{8}-0$

$[\cot \frac{\pi }{2}=0]$

$\Rightarrow \csc \frac{\pi }{32}+\csc \frac{\pi }{16}+\cot \frac{\pi }{16}$

$=\csc \frac{\pi }{32}+\cot \frac{\pi }{32}$

$=\cot \frac{\pi }{164}$

$RHS=\cot \frac{\pi }{K}$

On comparing we get $[K=64]$