Question

prove that L.H.S =R.H.S​

Answer 1

LHS = √sec © + 1/√sec © - 1 = (√sec © + 1)^2 / √sec^2 © - 1 = sec © + 1 / √tan^2 © = sec © + 1 / tan © = sec © / tan © + 1/tan © = cosec © + cot © = RHS

Hence, LHS = RHS
Hope this helps you !

Answer 2

Method:

$LHS \\ \\ \sqrt{\frac{\sec\theta+1}{\sec\theta-1}} \\ \\ \sqrt{\frac{(\sec\theta+1)(\sec\theta-1)}{(\sec\theta-1)(\sec\theta-1)}} \\ \\ \sqrt{\frac{\sec^2\theta-1}{(\sec\theta-1)^2}} \\ \\ \sqrt{\frac{\tan^2\theta}{(\sec\theta-1)^2}} \\ \\ \frac{\tan\theta}{\sec\theta-1} \\ \\ \frac{\tan\theta(\sec\theta+1)}{(\sec\theta-1)(\sec\theta+1)} \\ \\ \frac{\tan\theta(\sec\theta+1)}{\sec^2\theta-1} \\ \\ \frac{\tan\theta(\sec\theta+1)}{\tan^2\theta} \\ \\ \frac{\sec\theta+1}{\tan\theta}$

$\frac{\sec\theta}{\tan\theta}+\frac{1}{\tan\theta} \\ \\ \csc\theta+\cot\theta \\ \\ RHS$