Product To Sum Formulas Derivation

The product to sum formulae can be derived using the trigonometric sum/difference formulae. 

The Sum/difference formulae are given below:

• sin (A + B) = (½) {sin A cos B + cos A sin B}———— (1)
• sin (A – B) = (½) {sin A cos B – cos A sin B}———— (2)
• cos (A + B) = (½) {cos A cos B – sin A sin B}———— (3)
• cos (A – B) = (½) {cos A cos B + sin A sin B}———— (4)

cos A cos B Formula (Product of Cosines)

To derive the cos A cos B formula add equations (3) and (4)

⇒ cos (A + B) + cos (A – B) = [cos A cos B – sin A sin B] + [cos A cos B + sin A sin B]

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

⇒ cos (A + B) + cos (A – B) = 2 cos A cos B

Hence, 

  cos A cos B =  (½) [cos (A + B) + cos (A – B)]

cos A sin B Formula (Product of Sine and Cosine)

To derive the cos A sin B formula subtract equations (2) from (1)

⇒ sin (A + B) – sin (A – B) = [sin A cos B + cos A sin B] – [sin A cos B – cos A sin B]

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

⇒ sin (A + B) – sin (A – B) = 2 cos A sin B

Hence,

cos A sin B = (½) [sin (A + B) – sin (A – B)]

sin A cos B Formula (Product of Sine and Cosine)

To derive the sin A cos B formula add equations (1) and (2)

⇒ sin (A + B) + sin (A – B) = [sin A cos B + cos A sin B] + [sin A cos B – cos A sin B]

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

⇒ sin (A + B) + sin (A – B) = 2 sin A cos B

Hence,

sin A cos B = (½) [sin (A + B) + sin (A – B)]

sin A sin B Formula (Product of Sines)

To derive the sin A sin B formula subtract equations (4) from (3)

⇒ cos (A – B) – cos (A + B) = [cos A cos B + sin A sin B] – [cos A cos B – sin A sin B]

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

⇒ cos (A – B) – cos (A + B) = 2 sin A sin B

Hence,

sin A sin B = (½) [cos (A – B) – cos (A + B)]

Product-to-sum formulas are trigonometric identities that convert the product of sine and cosine functions into a sum (or difference) of trigonometric functions. These formulas are particularly useful in simplifying the integrals and solving trigonometric equations.

In this article, we will learn about, Product to Sum Formulae, related examples and others in detail.