Prove the following identity: cos x + cos 2x + cos 3x = cos 2x(1 + 2cos x)

We'll remove the brackets from the right
side:

cos x + cos 2x + cos 3x = cos 2x + 2cos x*cos
2x

We'll eliminate the term cos
2x:

cos x + cos 3x = 2cos x*cos
2x

We'll write cos 2x = 2(cos x)^2 -
1

cos 3x = 4(cos x)^3 - 3cos
x

We'll substitute them in identity and we'll
get:

cos x +  4(cos x)^3 - 3cos x = 2cos x*[2(cos x)^2 -
1]

We'll combine like terms from left side and we'll remove
the brackets form the right side:

4(cos x)^3
- 2cos x = 4(cos x)^3 - 2cos x q.e.d.