Question

Prove the following identities :
$\frac{1-\cos \theta }{1+\cos \theta }=(cosec\theta -\cot \theta {)}^2$

Answer 1

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

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

$\Rightarrow$ L.H.S. = $\frac{(1-\cos \theta {)}^2}{1-{\cos }^2\theta }=\frac{(1-\cos \theta {)}^2}{si{n}^2\theta }$

$\Rightarrow$ L.H.S. = ${\left(\frac{1-\cos \theta }{sin\theta }\right)}^2={(\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta })}^2=(cosec\theta -\cot \theta {)}^2=R.H.S.$