Starting with the formulas cosA B cos A cos B sin A sin B

Starting with the formulas cos(A + B) = cos A cos B sin A sin B and sin2 + cos2 = 1 derive the formula cos(2) = 2 cos2 1

Solution

cos(A + B) = cos A cos B sin A sin B

cos(2) = cos( + ) = cos cos sin sin = cos^2 - sin^2

Now sin^2 + cos^2 = 1 =====> sin^2 = 1 - cos^2

So, cos(2) = cos^2 - 1 + cos^2 = 2 cos^2 1 = RHS