Math Doubts

Difference to Product identities

In trigonometry, there are two difference to product transformation identities and they are used as formulas for expressing the difference of two trigonometric functions into their product form.

Sine functions

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

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

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

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

Cosine functions

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

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

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

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

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