Precalc: Prove the Identity: sin2x/1-cos2x = 2csc2x-tanx

Answer 1

We know the following: sin 2x = 2* sin x * cos x

cos 2x = 1-2*sin²x

Thus, we can plug these expression into the original left term and get:

left term = sin 2x / (1-cos 2x) = 2* sin x * cos x / (2*sin²x) = cos x / sin x

To tackle the right term, we use the identity: csc x = 1/sin x and then write:

right term = 2 * csc 2x - tan x = 2/ sin 2x - tan x

We plug in the identity for sin 2x just as we did above and then replace tan x and write:

right term = 2 * csc 2x - tan x = 2/ (2* sin x * cos x) = 1/ (sin x * cos x) - sin x / cos x = (1 - sin²x) / (sin x * cos x)

We notice we can replace : 1 - sin²x = cos²x

right term = cos²x/ (sin x * cos x) = cos x/ sin x

Thus, we have proved that the left term has the exact form as the right term, specifically cos x/ sin x :)