Sum to Product identities

A trigonometric identity that expresses the transformation of sum of the trigonometric functions into the product form of trigonometric functions is called the sum to product identity.

In trigonometry, there are two types of sum to product transformation identities and they are used as formulas in mathematics. Now, let’s learn the sum to product trigonometric identities with proofs.

Sine functions

$\sin{\alpha}+\sin{\beta}$ $\,=\,$ $2\sin{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\cos{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The sum of sine functions can be transformed into the product of the sine and cosine functions. It is called the sum to product transformation identity of the sine functions.

The sum to product identity of sine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\sin{x}+\sin{y}$ $\,=\,$ $2\sin{\Bigg(\dfrac{x+y}{2}\Bigg)}\cos{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\sin{C}+\sin{D}$ $\,=\,$ $2\sin{\Bigg(\dfrac{C+D}{2}\Bigg)}\cos{\Bigg(\dfrac{C-D}{2}\Bigg)}$

Cosine functions

$\cos{\alpha}+\cos{\beta}$ $\,=\,$ $2\cos{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\cos{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The sum of cosine functions can be transformed into the product of the cosine functions. It is called the sum to product transformation identity of the cosine functions.

The sum to product identity of cosine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\cos{x}+\cos{y}$ $\,=\,$ $2\cos{\Bigg(\dfrac{x+y}{2}\Bigg)}\cos{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\cos{C}+\cos{D}$ $\,=\,$ $2\cos{\Bigg(\dfrac{C+D}{2}\Bigg)}\cos{\Bigg(\dfrac{C-D}{2}\Bigg)}$