Write the difference in tanpi/4-cosx as a product?

First, we'll substitute the function tan pi/4 by it's
value 1.

To transform the difference into a product, we'll
have to express the value 1 as being the function cosine of an angle, so that the terms
of the difference to be 2 like trigonometric functions.

1 =
cos 0 or cos 2pi

1 - cosx = cos 0-cos
x

cos 0-cos x =  -2 sin (0+x)/2*sin
(0-x)/2

cos 0-cos x = -2sin (x/2)*sin
(-x/2)

Because of the fact that the trigonometric function
sine is an odd function , we'll write sin (-x/2)=-sin
(x/2)

cos 0-cos x = 2sin (x/2)*sin
(x/2)

tan pi/4 - cos x = 2[sin
(x/2)]^2