Question

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

Answer 1

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

Multiply and divide by $(1+\cos \theta )$

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

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

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

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

$=cosec\theta +\cot \theta$

$=RHS$

Hence proved