Prove the identity algebraically? Help please.

Cos3x= 4cos^3x - 3cosx

I tried forever! But i cant seem to solve this identity......

cos(3x)

= cos(2x + x)

= cos(2x) cos(x) - sin(2x) sin(x)

= cos x (2 cos^2(x) - 1) - sinx (2 sin x cos x)

= 2cos^3(x) - cos x - 2sin^2(x) cos x

= 2cos^3(x) - cos x - 2 cos x (1 - cos^2(x))

= 2cos^3(x) - cos x - 2cos x + 2cos^3(x)

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

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

Simplifying the LHS:

cos(3x) = cos(2x + x) = cos(2x)*cos(x) - sin(2x)*sin(x)

= (cos²(x) - sin²(x))*cos(x) - 2*sin²(x)*cos(x)

= cos³(x) - 3sin²(x)cos(x)

= cos³(x) - 3cos(x)(1 - cos²(x))

= 4cos³(x) - 3cos(x)