Prove that (sin A + cos A)/(sin A – cos A) + (sin A – cos A)/(sin A + cos A) = 2/(2 sin2 A – 1)

Question

Prove that $\frac{sin A + cos A}{sin A – cos A} + \frac{sin A – cos A}{sin A + cos A} = \frac{2}{2 sin2 A – 1}$

Sapling Academy · Accepted Answer

Let's start solving from LHS: $\frac{sin A + cos A}{sin A – cos A} + \frac{sin A – cos A}{sin A + cos A}$ = $\frac{(sin A + cos A)(sin A + cos A) + (sin A – cos A)(sin A – cos A)}{(sin A – cos A)(sin A + cos A)}$ = $\frac{(sin ^2 A + cos ^2 A + 2 sin A cos A) + (sin ^2 A + cos ^2 A - 2 sin A cos A)}{(sin^2 A – cos^2 A)}$ = $\frac{(sin ^2 A + cos ^2 A + 2 sin A cos A + sin ^2 A + cos ^2 A - 2 sin A cos A)}{(sin^2 A – cos^2 A)}$ = $\frac{(sin ^2 A + cos ^2 A + \cancel{2 sin A cos A} + sin ^2 A + cos ^2 A - \cancel{2 sin A cos A})}{(sin^2 A – cos^2 A)}$ = $\frac{2 (sin ^2 A + cos ^2 A)}{(sin^2 A – cos ^2 A)}$ ∵ sin2 A + cos2 A = 1, ∴ cos2 A = 1 - sin2 A ∴ LHS =  $\frac{2 (1)}{(sin^2 A – (1 - sin ^2 A)}$ = $\frac{2 (1)}{(sin^2 A – 1 + sin ^2 A)}$ = $\frac{2}{2 sin^2 A – 1}$ = RHS....  Hence Proved ! Please press the "Heart", if you like the solution.

📚 Get Free Worksheet & Share Solution

Leave a Comment Cancel Reply

Contact us

Join us

Useful Link

Our Quizzes

📚 Get Free Worksheet & Share Solution

Related Math Questions

Leave a Comment Cancel Reply