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• Trigonometry
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Sum to Product identities

A trigonometric identity to transform sum or difference of angle sum and angle difference identities as product form of trigonometric functions is called sum (or) difference to product identity.

There are four popular sum (or) difference to product identities and they are used as formulas in trigonometry. Each sum (or) difference to product identity is popularly expressed mathematically in three forms. You can follow any one of them.

Sine of Sum to Product Identity

The sum of sines of angle sum and difference identities to product form transformation is called sine of sum to product identity.

$(1) \,\,\,$ $\sin{(A+B)}$ $+$ $\sin{(A-B)}$ $\,=\,$ $2\sin{A}\cos{B}$

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

$(3) \,\,\,$ $\sin{(\alpha+\beta)}$ $+$ $\sin{(\alpha-\beta)}$ $\,=\,$ $2\sin{\alpha}\cos{\beta}$

Sine of Difference to Product Identity

The subtraction (or difference) of sines of angle sum and difference identities to product form transformation is called sine of difference to product identity.

$(1) \,\,\,$ $\sin{(A+B)}$ $-$ $\sin{(A-B)}$ $\,=\,$ $2\cos{A}\sin{B}$

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

$(3) \,\,\,$ $\sin{(\alpha+\beta)}$ $-$ $\sin{(\alpha-\beta)}$ $\,=\,$ $2\cos{\alpha}\sin{\beta}$

Cosine of Sum to Product Identity

The sum of cosines of angle sum and difference identities to product form transformation is called cosine of sum to product identity.

$(1) \,\,\,$ $\cos{(A+B)}$ $+$ $\cos{(A-B)}$ $\,=\,$ $2\cos{A}\cos{B}$

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

$(3) \,\,\,$ $\cos{(\alpha+\beta)}$ $+$ $\cos{(\alpha-\beta)}$ $\,=\,$ $2\cos{\alpha}\cos{\beta}$

Cosine of Difference to Product Identity

The subtraction (or difference) of cosines of angle sum and difference identities to product form transformation is called sine of difference to product identity.

$(1) \,\,\,$ $\cos{(A+B)}$ $-$ $\cos{(A-B)}$ $\,=\,$ $-2\sin{A}\sin{B}$

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

$(3) \,\,\,$ $\cos{(\alpha+\beta)}$ $-$ $\cos{(\alpha-\beta)}$ $\,=\,$ $-2\sin{\alpha}\sin{\beta}$

Other form

$\sin{C}$ $+$ $\sin{D}$ $\,=\,$ $2\sin{\Bigg[\dfrac{C+D}{2}\Bigg]}\cos{\Bigg[\dfrac{C-D}{2}\Bigg]}$

$\sin{C}$ $-$ $\sin{D}$ $\,=\,$ $2\cos{\Bigg[\dfrac{C+D}{2}\Bigg]}\sin{\Bigg[\dfrac{C-D}{2}\Bigg]}$

$\cos{C}$ $+$ $\cos{D}$ $\,=\,$ $2\cos{\Bigg[\dfrac{C+D}{2}\Bigg]}\cos{\Bigg[\dfrac{C-D}{2}\Bigg]}$

$\cos{C}$ $-$ $\cos{D}$ $\,=\,$ $-2\sin{\Bigg[\dfrac{C+D}{2}\Bigg]}\sin{\Bigg[\dfrac{C-D}{2}\Bigg]}$