Question

Prove that: $\frac{1}{\sec \theta -1}-\frac{1}{\sec \theta +1}=2{\cot }^2\theta$

Answer 1

To prove: $\frac{1}{\sec \theta -1}-\frac{1}{\sec \theta +1}=2{\cot }^2\theta$

Lets take LHS and then equate it to RHS.

$LHS=\frac{1}{\text{sec}\theta -1}-\frac{1}{\text{sec}\theta +1}$

$=\frac{(\text{sec}\theta +1)-(\text{sec}\theta -1)}{(\text{sec}\theta -1)(\text{sec}\theta +1)}$

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

$=\frac{2}{{\tan }^2\theta }$

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

$=RHS$

$\Rightarrow L.H.S=R.H.S$

$\therefore \frac{1}{\sec \theta -1}-\frac{1}{\sec \theta +1}=2{\cot }^2\theta$
Hence, Proved