QJEEAdv11Ma36d8: JEE Advanced 2011 Mathematics - Class 11 JEEAdv11Ma36d8 Solution: The positive integer value of satisfying the equation is

We have,

${1 \over {\sin (\pi /n)}} - {1 \over {\sin (3\pi /n)}} = {1 \over {\sin (2\pi /n)}}$

$\Rightarrow {{\sin (3\pi /n) - \sin (\pi /n)} \over {\sin (\pi /n)\sin (3\pi /n)}} = {1 \over {\sin (2\pi /n)}}{{(2\sin (\pi /n)\cos (2\pi /n))\sin (2\pi /n)} \over {\sin (\pi /n)\sin (3\pi /n)}} = 1$

$\Rightarrow \sin {{4\pi } \over n} = \sin {{3\pi } \over n} \Rightarrow {{4\pi } \over n} + {{3\pi } \over n} = \pi \Rightarrow n = 7$