Sum or Difference to Product Identities

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

Introduction

In mathematics, the trigonometric functions are involved in fundamental operations like addition or subtraction. If the angles in the trigonometric functions are same, we know how to add or subtract them directly but we cannot add or subtraction them directly if the angles in the trigonometric functions are different.

Hence, we should use the following transformation trigonometric identities to add or subtract trigonometric functions in mathematics.

Sum to Product

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

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

Difference to Product

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

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