Double Angle Formulas of Cos

The sum formula of cosine function is,

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

When A = B, the above formula becomes,

cos (A + A) = cos A cos A – sin A sin A

cos 2A = cos²A – sin²A

Let us use this as a base formula to derive two other formulas of cos 2A using the Pythagorean identity sin²A + cos²A = 1.

(i) cos 2A = cos²A − (1 − cos²A) = 2cos²A – 1

(ii) cos 2A = (1- sin²A) – sin²A = 1 – 2sin²A

Now, we will derive the formula of cos 2A in terms of tan using the base formula.

cos2A=cos2A−sin2A=cos2A(1−sin2Acos2A)=1sec2A(1−tan2A)=11+tan2A(1−tan2A)=1−tan2A1+tan2Acos⁡2A=cos2⁡A−sin2⁡A=cos2⁡A(1−sin2⁡Acos2⁡A)=1sec2⁡A(1−tan2⁡A)=11+tan2⁡A(1−tan2⁡A)=1−tan2⁡A1+tan2⁡A

Thus, the double angle formulas of the cosine function are:

cos 2A = cos²A – sin²A (or) 2cos²A – 1 (or) 1 – 2sin²A (or) (1 – tan²A) / (1 + tan²A)