Question

If $\cos \theta =-\frac{3}{5},{\pi }^c<\theta <\frac{3{\pi }^c}{2}$, find the value of $\frac{\csc \theta +\cot \theta }{\sec \theta -\tan \theta }$

Answer 1

Given,

$\frac{\csc \theta +\cot \theta }{\sec \theta -\tan \theta }$

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

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

Given, $\cos \theta =-\frac{3}{5}$

$\Rightarrow \sin \theta =\frac{4}{5}$

$=\frac{\frac{1-\frac{3}{5}}{\frac{4}{5}}}{\frac{1-\frac{4}{5}}{-\frac{3}{5}}}$

$=-\frac{6}{4}=-\frac{3}{2}$