Question

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

Answer 1

To Prove:

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

Solution:

L.H.S $=\sqrt{\frac{1+\cos \theta }{1-\cos \theta }}$

Multiplying $(1+\cos \theta )$ with numerator and denominator both we get,

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

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

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

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

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

$=\text{cosec}\theta +\cot \theta$

$=$ R.H.S

Hence, L.H.S $=$ R.H.S