Difference to Product identities

A trigonometric identity that expresses the transformation of difference of the trigonometric functions into the product form of trigonometric functions is called the difference to product identity.

There are two types of difference to product transformation identities and they are used as formulas in trigonometry. Now, let us learn the difference to product trigonometric identities with proofs.

Sine functions

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

The difference of sine functions can be transformed into the product of the cosine and sine functions. It is called the difference to product transformation identity of the sine functions.

The difference to product identity of sine functions is also written in the following two forms popularly.

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

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

Cosine functions

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

The difference of cosine functions can be transformed into the product of the sine functions. It is called the difference to product transformation identity of the cosine functions.

The difference to product identity of cosine functions is also written in the following two forms popularly.

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

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

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