Complete the identity. sin4x - cos4x = ?

Since sin (2x) = 2sinxcosx, we can plug that in to get sin (4x) = 2sin (2x) cos (2x) = 2*2sinxcosxcos (2x) = 4sinxcosxcos (2x). Since cos (2x) = cos^2x-sin^2x, we plug that in. In addition, cos4x=cos^2 (2x) - sin^2 (2x). Next, since cos^2x = (1+cos (2x)) / 2 and sin^2x = (1-cos (2x)) / 2, we plug those in to end up with 4sinxcosxcos (2x) - ((1+cos (2x)) / 2 - (1-cos (2x)) / 2)

=4sinxcosxcos (2x) - (2cos (2x) / 2) = 4sinxcosxcos (2x) - cos (2x)

=cos (2x) * (4sinxcosx-1). Since sinxcosx=sin (2x), we plug that back in to end up with cos (2x) * (4sin (2x) - 1)