Question

Solve : $\frac{1}{\text{cosec}\theta +\cot \theta }-\frac{1}{\sin \theta }=\frac{1}{\sin \theta }-\frac{1}{\text{cosec}\theta -\cot \theta }$

Answer 1

$\frac{1}{\csc \theta +\cot \theta }-\frac{1}{\sin \theta }=\frac{1}{\sin \theta }-\frac{1}{\csc \theta -\cot \theta }$

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

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

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

$2\frac{\sin \theta }{1-{\cos }^2\theta }=2\frac{1}{\sin \theta }$

$\frac{\sin \theta }{{\sin }^2\theta }=\frac{1}{\sin \theta }$

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

$1=1$