tan^2(5x)cos^4(5x) = 1/8 - 1/8cos(20x)

Answer 1

tan^2(5x)cos^4(5x) = 1/8 - 1/8cos(20x)

since tan = sin/cos, you have
sin^2(5x)cos^2(5x) = 1/8 - 1/8cos(20x)

Now use the double-angle formula on the left, and the half-angle formula on the right to get

1/4 sin^2(10x) = 1/4 sin^2(10x)
QED