Math Doubts

Sum to Product identities

There are two sum to product transformation identities in trigonometry and they are used as formulas for expressing the sum of two trigonometric functions into their product form.

Sine functions

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

It is called the sum to product transformation identity of sine functions and it is also popularly written in the following two forms.

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

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

Cosine functions

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

It is called the sum to product transformation rule of cosine functions and it is also written in the following two forms popularly.

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

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

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