how to solve using identities?

Question

2sin(3x)cos(3x)(2cos^2 3x-1)

Answer 1

You can't solve this because it isn't an equation--there's no equal sign! You could try to simplify it.

If you simplify it, you should get (1/2) sin(12x)

Just to be clear, I assume we are simplifying 2sin(3x)cos(3x)[2cos²(3x) - 1].

We are going to use the double angle identities:

sin(2θ) = 2sinθcosθ

cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

Going from left to right, if we look at 2sin(3x)cos(3x), we can apply the double angle identity for sine and get sin(2*3x) = sin(6x).

After that, we can apply the double angle identity for cosine and get 2cos²(3x) - 1 = cos(2*3x) = cos(6x)

So we end up with sin(6x)cos(6x). From here we can apply the double angle identity for sine again:

sin(6x)cos(6x) = (1/2) * 2sin(6x)cos(6x) = (1/2) sin(2*6x) = (1/2) sin(12x).