Proving Trigonometric Identities Prove: cos3A = -3cosA + 4cos^3A

We need to prove that:

cos3A
= -3cosA + 4cos^3 A

We will start from the left sides and
prove the right sides.

We know
that:

cos3A = cos(2A + A)

Now
we will use the trigonometric identities to prove.

We know
that:

cos(a+b) = cosa*cosb -
sina*sinb

==> cos(2A+A) = cos2A*cosA -
sin2A*sinA

Now we know
that:

cos2A = 2cos^2 A
-1

sin2A =
2sinA*cosA

==> cos(2A+A) = (2cos^2 A -1)cosA -
2sinA*cosA*sinA

  Let us
simplify:

==> cos(2A+A) = 2cos^3 A - cosA - 2sin^2 A
* cosA

Now we know that: sin^2 A = 1- cos^2
A

==> cos(2A+A) = 2cos^3 A - cosA -2(1-cos^2
A)*cosA

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

Now we will combine like
terms.

==> cos(2A+A) = 4cos^3 A -
3cosA

==> cos(3A) = -3cosA + 4cos^3
A......q.e.d