Math Doubts

Sum to Product identity of Cosine functions


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

An identity that expresses the transformation of sum of cosine functions into product form is called the sum to product identity of cosine functions.


When $\alpha$ and $\beta$ represent two angles of the right triangles. The cosine of the two angles are written as $\cos{\alpha}$ and $\cos{\beta}$ in trigonometric mathematics. The sum of the two cosine functions is written mathematically as follows.


The sum of cosine functions can be transformed into the product of the trigonometric functions in the following mathematical form.

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

Other forms

The sum to product transformation rule of cosine functions is also popularly written in the following two forms in mathematics.

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

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

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


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

Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more