20 ! trig proof sin²(x)cos²x)=(1-cos(4x))/8

refresh if you see display errors such as \begin{aligned}, etc.focusing on the right side since it looks more complex:     if we rewrite  as  , then we can treat  as the angle that is being doubled. that will allow us to use the cosine double angle identity:       but notice how next to  , we have a 1. if we want to get rid of that 1, we can choose another version of the cosine double angle identity:     so this leads us to    we have a sine double angle identity:   in order to use it, we need to get rid of that square.since  is the same as  , which is the same as  , we have    use the sine double angle identity twice.    multiplication order does not matter. multiply all the sines together. multiply all the cosines together. multiply all the 2s together to form 8, which will cancel with denominator.