sin2x/1-sin^x = 2tanx

Question

prove the identity above

Answer 1

To prove this, we need only work with the left side of the equation. We begin by replacing sin 2x in the numerator with its equivalent expression 2 sin x cos x ( via the double-angle identity for sine). We then replace the denominator with cos² x ( via the first of the three Pythagorean Identities). We then cancel out the common factors of cos x in numerator and denominator, which leaves us with 2 sinx/cos x, and this is equivalent to 2 tan x, which now matches the right side of the equation.

Answer 2

sin2x/(1-sinx**2) = 2tanx

sin2x = 2 sinx cosx (double angle formula)

Substituting:

2 sinx cosx

--------------- = 2tanx

1-sinx**2

sinx**2 + cosx**2 = 1 --> cosx**2 = 1 - sinx**2

Substituting:

2sinx cosx

-------------- = 2tanx

cosx**2

Cancel cosx from numerator and denonimator:

2sinx

------- = 2tanx

cosx

2tanx = 2tanx --- done