Question

Solve for $\alpha$

$\frac{{\text{cosec}}^2\theta }{\text{cosec}\theta -1}-\frac{{\text{cosec}}^2\theta }{\text{cosec}\theta +1}=\alpha {\sec }^2\theta$

Answer 1

$\frac{{\csc }^2\theta }{\csc \theta -1}-\frac{{\csc }^2\theta }{\csc \theta +1}$

$=\frac{(\csc \theta +1){\csc }^2\theta -(\csc \theta -1){\csc }^2\theta }{{\csc }^2\theta -1}$

$=\frac{{\csc }^2\theta (\csc \theta +1-\csc \theta +1)}{{\csc }^2\theta -1}$

$=\frac{2{\csc }^2\theta }{{\csc }^2\theta -1}$

$=2{\sec }^2\theta$

$\Rightarrow 2{\sec }^2\theta =\alpha {\sec }^2\theta$

$\Rightarrow \alpha =2$