Question

Prove the identity $\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=cosec\theta -\cot \theta$

Answer 1

Consider $\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\sqrt{\frac{(1-\cos \theta )}{(1+\cos \theta )}\times \frac{(1-\cos \theta )}{(1-\cos \theta )}}$

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

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

$=cosec\theta -\cot \theta$