8 sin²(3x) cos²(3x) = 2 [2 sin(3x) cos(3x)]²

Question

= 2 [sin(6x)]²
= 2[1 - cos(12x)]/2
= 1 - cos(12x)

Answer 1

good job

I usually pick some angle, say x = 21,
and sub it into the original, and the final.
of course they have to be the same.

This does of course not "prove" your answer is right, but the probability that you would get the same result if you had an error would be "very small"

Of course if you don't get the same result, then you know you are wrong.