Prove the identity (sin2x)/(1+(tanx)^2) = 2sinx(cosx)^3
For a proof like this, I like to start with the more difficult-looking side, and aim to end up with the expression on the other side. Starting with the left hand side, the first thing to change is the sin2x, because the expression on the RHS only has terms of x, not 2x.Using the identity sin2x = 2sinxcosx, the expression becomes 2sinxcosx/(1 + (tan2x)).Next, we will change the tan2x to (sinx/cosx)2. This will give a fraction on the demoninator, which we will want to get rid of, so the next step is to multiply top and bottom of the expression by cos2x.This will give us 2sinxcosx.cos2x/cos2x+sin2x. Using the identity cos2x +sin2x = 1, and tidying up the numerator, we get to the required result of 2sinx(cos3x).
