Question

Solve it

$cot \frac{\theta}{2} - cot {\theta} = cosec \frac{\theta}{2}$

Answer 1

$\bold{Thanks\:For Asking\:Question}$

$\frac{cos \frac{ \theta}{2} }{sin \frac{ \theta}{2} } - \frac{cos \theta}{sin \theta} = \frac{1}{sin \frac{ \theta}{2} }$

$\frac{sin \theta \: cos \frac{ \theta}{2} - cos \theta \: sin \frac{ \theta}{2} }{sin \frac{ \theta}{2}.sin \theta } = \frac{1}{sin \frac{ \theta}{2} }$

$sin \frac{ \theta}{2} sin( \theta - \frac{ \theta}{2} ) = sin \frac{\theta}{2} sin{\theta} \\ \\ {sin}^{2} \frac{\theta}{2} - sin \frac{\theta}{2} sin{\theta} = 0$

${sin}^{2} \frac{\theta}{2} (1 - 2cos \frac{\theta}{2} ) = 0 \\ \\ sin {}^{2} \frac{\theta}{2} = 0 = sin {}^{2} 0 \\ \\ {sin}^{2} \frac{\theta}{2} = {sin}^{2} (n\pi { + -0) } \\ \\{ \theta} = 2n\pi$

$1 - 2cos \frac{\theta}{2} = 0 \\ 2cos \frac{\theta}{2} = \frac{1}{2} \: \: \: \: \: \: = cos\frac{\pi}{3} \\ \\ cos \frac{\theta}{2} = cos(2n\pi { + }{ - } \frac{\pi}{3} \\ \\ {\theta} = 4n\pi + - \frac{2\pi}{3}$

$\underline{Hence\:solved}$

$\bold{Swarnimkumar21}$