cos 3A is equal to

We know that $$cos (x + y) = cos x . cos y - sin x . sin y$$

Also, $$cos 2x = 2 cos^2 x - 1$$ and $$sin 2x = 2 sinx . cos x$$

Expression : $$cos 3A = cos (2A + A)$$

= $$cos 2A . cos A - sin 2A . sin A$$

= $$(2 cos^2 A - 1) cos A - (2 sin A . cos A) sin A$$

= $$2 cos^3 A - cos A - 2 sin^2 A . cos A$$

= $$2 cos^3 A - cos A - 2 cos A (1 - cos^2 A)$$

= $$2 cos^3 A - cos A - 2 cos A + 2 cos^3 A$$

= $$4 cos^3 A - 3 cos A$$