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.

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

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

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

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

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

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

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

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

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

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

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

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

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