tan2A + sin2A = 4tanA/1-tan⁴A | Prove the trigonometric identity | SciPiPupil

Question: Prove the following trigonometric identity: tan2A +sin2A = $\dfrac{4tanA}{1-tan⁴A}$

Solution:

Given,

LHS

= tan 2A + sin 2A

$\dfrac{2 tanA}{1 -tan²A} + \dfrac{2 tanA}{1 +tan²A}$

$\dfrac{2 tanA(1 +tan²A) + 2tanA (1 -tan²A)}{(1 -tan²A)(1 +tan²A)}$

$\dfrac{2tanA (1 +tan²A +1 -tan²A}{1 -tan⁴A}$

$\dfrac{2tanA (2)}{1 -tan⁴A}$

$\dfrac{4tanA}{1 -tan⁴A}$

RHS

tan2A + sin2A = 4tanA/1-tan⁴A | Prove the trigonometric identity | SciPiPupil