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

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

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

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

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.