Question

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

Answer 1

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.
$\frac{2 sin(3x) cos(3x)}{1+(2cos^2 (3x) -1)} = \frac{2 sin(3x) cos(3x)}{2cos^2 (3x) } = \frac{sin(3x)}{cos(3x)} = tan (3x)$