I already answered the first one in the other thread.

I would think of the second one with the replacement y = x/2. (I hate 1/2 angle problems!) That transforms the problem to showing:

tan(y) = sin(2y) / [1 + cos(2y)]

The RHS is

sin(2y) = 2sin(y)cos(y)

cos(2y) = 2[cos(y)]^2 - 1

So 1 + cos(2y) = 1 + 2[cos(y)]^2 - 1 = 2[cos(y)]^2

Dividing:

sin(2y) / [1 + cos(2y)] = [2sin(y)cos(y)]/[2{cos(y)}^2] = sin(y)/cos(y) = tan(y).

- Dan