Question

Show that $\frac{1}{\text{cosec}\theta -\cot \theta }-\frac{1}{\sin \theta }=\frac{1}{\sin \theta }-\frac{1}{\text{cosec}\theta +\cot \theta }$

Answer 1

We know, ${\csc }^2\theta -{\cot }^2\theta =1$

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

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

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

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

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