Question

Solve :
$(cosec\theta -cot\theta )\sqrt{\frac{1+cos\theta }{1-cos\theta }}=1$

Answer 1

$\text{cosec}\theta -\cot \theta \sqrt{\frac{1+\cos \theta }{1-\cos \theta }}=1$

$=(\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta })\left(\sqrt{\frac{\frac{2{\cos }^2\theta }{2}}{\frac{2{\sin }^2\theta }{2}}}\right)$

$=\left(\frac{1-\cos \theta }{\sin \theta }\right)\left(\frac{\cot \theta }{2}\right)$

$=\frac{\frac{2{\sin }^2\theta }{2}}{\frac{2\sin \theta }{2}\times \frac{\cos \theta }{2}}\times \frac{cot\theta }{2}$

$=\frac{\tan \theta }{2}\times \frac{cot\theta }{2}$

$=1$ (proved)