Transformation identities

The trigonometric functions are often required to transform from sum to product and product to sum form in mathematics. Hence, some transformation formulas are derived in trigonometry to do it easily.

01

Sum to Product formulas

There are two types of sum to product transformation rules in trigonometry.

Compound angles

The following trigonometric laws are used when sum of trigonometric functions appears with compound angles and required to transform them as product.

$(1)\,\,\,\,$ $\sin{(a+b)}+\sin{(a-b)}$ $\,=\,$ $2\sin{a}\cos{b}$

$(2)\,\,\,\,$ $\sin{(a+b)}-\sin{(a-b)}$ $\,=\,$ $2\cos{a}\sin{b}$

$(3)\,\,\,\,$ $\cos{(a+b)}+\cos{(a-b)}$ $\,=\,$ $2\cos{a}\cos{b}$

$(4)\,\,\,\,$ $\cos{(a+b)}-\cos{(a-b)}$ $\,=\,$ $-2\sin{a}\sin{b}$

Angles

The following trigonometric identities are used when sum of trigonometric functions appear with angles and required to transform them in product form of the trigonometric functions.

$(1)\,\,\,\,$ $\sin{c}+\sin{d}$ $\,=\,$ $2\sin{\Bigg[\dfrac{c+d}{2}\Bigg]}\cos{\Bigg[\dfrac{c-d}{2}\Bigg]}$

$(2)\,\,\,\,$ $\sin{c}-\sin{d}$ $\,=\,$ $2\cos{\Bigg[\dfrac{c+d}{2}\Bigg]}\sin{\Bigg[\dfrac{c-d}{2}\Bigg]}$

$(3)\,\,\,\,$ $\cos{c}+\cos{d}$ $\,=\,$ $2\cos{\Bigg[\dfrac{c+d}{2}\Bigg]}\cos{\Bigg[\dfrac{c-d}{2}\Bigg]}$

$(4)\,\,\,\,$ $\cos{c}-\cos{d}$ $\,=\,$ $-2\sin{\Bigg[\dfrac{c+d}{2}\Bigg]}\sin{\Bigg[\dfrac{c-d}{2}\Bigg]}$

02

Product to Sum formulas

The below four transformation rules are used if trigonometric functions in various combinations are in product form and require to convert them in sum of the trigonometric functions.

$(1)\,\,\,\,$ $\sin{a}\cos{b}$ $\,=\,$ $\dfrac{[\sin{(a+b)}+\sin{(a-b)}]}{2}$

$(2)\,\,\,\,$ $\cos{a}\sin{b}$ $\,=\,$ $\dfrac{[\sin{(a+b)}-\sin{(a-b)}]}{2}$

$(3)\,\,\,\,$ $\cos{a}\cos{b}$ $\,=\,$ $\dfrac{[\cos{(a+b)}+\cos{(a-b)}]}{2}$

$(4)\,\,\,\,$ $\sin{a}\sin{b}$ $\,=\,$ $\dfrac{[\cos{(a-b)}-\sin{(a+b)}]}{2}$

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