Question

Prove that
${(\cot \frac{\theta }{2}-\tan \frac{\theta }{2})}^2(1-2\tan \theta \cot 2\theta )=$

Answer 1

${(\cot \frac{\theta }{2}-\tan \frac{\theta }{2})}^2[1-2\tan \theta \frac{1}{\tan 2\theta }]$

$={(\cot \frac{\theta }{2}-\tan \frac{\theta }{2})}^2[1-\frac{2\tan \theta }{2\tan \theta }\times (1-{\tan }^2\theta \left)\right]$

$=({\cot }^2\frac{\theta }{2}+{\tan }^2\frac{\theta }{2}-2)\left({\tan }^2\theta \right)$

$=({\cot }^2\frac{\theta }{2}+{\tan }^2\frac{\theta }{2}-2){\left(\frac{2\tan \theta /2}{1-{\sin }^2\theta /2}\right)}^2$

$=\frac{(4+4{\tan }^4\frac{\theta }{2}-8{\tan }^2\frac{\theta }{2})}{{(1-{\tan }^2\frac{\theta }{2})}^2}$

$=\frac{4(1+{\tan }^4\frac{\theta }{2}-2{\tan }^2\frac{\theta }{2})}{{(1-{\tan }^2\frac{\theta }{2})}^2}$

$=4$