# Angle difference formulas

A trigonometric identity to expand a trigonometric function having difference of two angles is called the angle difference identity. In trigonometry, there are four angle difference trigonometric identities and they’re used as formulas in mathematics. Let’s start to study all the angle difference identities with proofs.

### Sine angle difference 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 difference 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 difference 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 difference formula

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

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

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

Email subscription
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more