$\frac{1}{\operatorname{cosec} heta(1-\cot heta)}+\frac{1}{\sec heta(1- an heta)}$ is equal

Question

$\frac{1}{\operatorname{cosec} 	heta(1-\cot 	heta)}+\frac{1}{\sec 	heta(1-	an 	heta)}$ is equal to\

Professor PC Thomas Classes · Answer

(c) We have, $\frac{1}{\operatorname{cosec} 	heta(1-\cot 	heta)}+\frac{1}{\sec 	heta(1-	an 	heta)}$ $$ \begin{aligned} & =\frac{\sin 	heta}{1-\frac{\cos 	heta}{\sin 	heta}}+\frac{\cos 	heta}{1-\frac{\sin 	heta}{\cos 	heta}} \ & {\left[\because \frac{1}{\operatorname{cosec} 	heta}=\sin 	heta, \frac{1}{\sec 	heta}=\cos 	heta, 	an 	heta=\frac{\sin 	heta}{\cos 	heta}, \cot 	heta=\frac{\cos 	heta}{\sin 	heta}ight]} \ & =\frac{\sin ^{2} 	heta}{\sin 	heta-\cos 	heta}+\frac{\cos ^{2} 	heta}{\cos 	heta-\sin 	heta}=\frac{\sin ^{2} 	heta-\cos ^{2} 	heta}{\sin 	heta-\cos 	heta}=\sin 	heta+\cos 	heta \end{aligned} $$