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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