# Difference to Product identity of Sine functions

## Formula

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

An identity that expresses the transformation of difference of sine functions into product form is called the difference to product identity of sine functions.

### Introduction

When the $\alpha$ and $\beta$ represent the two angles of right triangles, the sine functions with both angles are written as $\sin{\alpha}$ and $\sin{\beta}$ in mathematical form. The difference of the two sine functions is written in the following mathematical form in trigonometry.

$\sin{\alpha}-\sin{\beta}$

The difference of sine functions can be transformed into the product of trigonometric functions as follows.

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

#### Other forms

The difference to product transformation formula for the sin functions is written in two popular forms.

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

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

Thus, we can write the difference to product transformation formula for sine functions in terms of any two angles.

#### Proof

Learn how to derive the difference to product transformation identity of sine functions in mathematics.

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