Prove the identity (sin6x)/(1+cos6x)=tan3x.

Use double angle identity

$sin(2x) = 2 sin x cos x \\ \\ cos(2x) = 2cos^2 x - 1$

We can relate '6x' to '3x' in the same way since 6 is 2*3.

$sin(6x) = 2 sin (3x) cos (3x) \\ \\ cos(6x) = 2cos^2 (3x) - 1$

Now sub into left side of identity. Simplify until it equals right side.

(2 sin(3x) cos(3x))/(1+(2cos^2 (3x) -1)) = (2 sin(3x) cos(3x))/(2cos^2 (3x) ) = (sin(3x))/(cos(3x)) = tan (3x)