Question

Prove that $\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=cosec\theta -\cot \theta$.

Answer 1

LHS $=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}$

Multiply and divide by LHS by $\left(\sqrt{1-\cos \theta }\right)$

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

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

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

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

$=cosec\theta -\cot \theta$

$=RHS$

Hence proved