A trigonometric identity that expresses the relation between a trigonometric function with sum of angles and the trigonometric functions with both angles is called the angle sum trigonometric identity. In trigonometry, the following are some of the angle sum formulas with proofs, uses and problems with solutions.

The sine of the sum of two angles is equal to the sum of the products of the sines and cosines of both angles.

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

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

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

The cosine of the sum of two angles is equal to the subtraction of the product of sines of both angles from the product of cosines of both angles.

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

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

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

The tangent of the sum of two angles is equal to the quotient of the sum of the tangents of both angles by the subtraction of the product of tangents of both angles from one.

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

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

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

The cotangent of the sum of two angles is equal to the quotient of the subtraction of one from the product of cotangents of both angles by the sum of the cotangents of both angles.

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

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

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

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