# Difference to Product identity of Sin functions

## Formula

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

The transformation of difference of two sin functions into product form is called the difference to product identity of sin functions.

### Introduction

Let $\alpha$ and $\beta$ represent two angles of two right triangles, the sine functions with the same angles are written in mathematical form as $\sin{\alpha}$ and $\sin{\beta}$. In trigonometry, the sine functions participate in subtraction in some cases and the difference of them is expressed in mathematical form as follows.

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

The subtraction of the sine functions can be transformed into product form for simplifying it.

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

#### Popular forms

The difference to product transformation rule of sin functions is also popularly written in two other forms.

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

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

In this way, you can express the difference to product formula of sine functions in terms of any two angles.

#### Proof

Learn how to derive the difference to product transformation identity of sin functions in trigonometry.

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