Andra M. answered • 11/27/22
Ivy League Tutor and mentor (Columbia BA, NYU PhD)
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 :)
Andra M.
Thanks Mark! Updating the answer accordingly: Once we have the left term as cos x/ sin x, we will use the identifies listed, first aiming to get to an expression with csc 2x. To do that, we write: left term = cos x/ sin x = 2 cos^2 x/ sin 2x = 2 (1- sin^2 x) / sin 2x = 2/ sin 2x - tan x =2 csc 2x - tan x11/27/22
Mark M.
Verification of identities are usually done by transforming only one side of the identity.11/27/22