Angle sum formulas

A trigonometric identity to expand a trigonometric function having sum of two or more angles is called the angle sum identity. There are four popularly used angle sum trigonometric identities in trigonometry. Let’s start to learn all the angle sum identities with proofs.

Sine angle sum formula

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

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

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

Cosine angle sum formula

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

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

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

Tangent angle sum formula

$(1) \,\,\,\,$ $\tan{(A+B)}$ $\,=\,$ $\dfrac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}$

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

$(3) \,\,\,\,$ $\tan{(\alpha+\beta)}$ $\,=\,$ $\dfrac{\tan{\alpha}+\tan{\beta}}{1-\tan{\alpha}\tan{\beta}}$

Cotangent angle sum formula

$(1) \,\,\,\,$ $\cot{(A+B)}$ $\,=\,$ $\dfrac{\cot{B}\cot{A}-1}{\cot{B}+\cot{A}}$

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

$(3) \,\,\,\,$ $\cot{(\alpha+\beta)}$ $\,=\,$ $\dfrac{\cot{\beta}\cot{\alpha}-1}{\cot{\beta}+\cot{\alpha}}$

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