Math Doubts

Difference to Product identity of Cos functions

Formula

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

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

Introduction

Assume, $\alpha$ and $\beta$ as the angles of two right triangles. The cosines of the both angles are written in mathematical form as $\cos{\alpha}$ and $\cos{\beta}$ in trigonometry. The subtraction of the two cosine functions are expressed in mathematical form as follows.

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

The cos functions are often involved in subtraction in some expressions and equations. So, the difference of the cos functions can be transformed into product form as follows.

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

Popular forms

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

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

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

Thus, you can write the difference to product formula of cosine functions in terms of any two angles.

Proof

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

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