Compound angle formulas

Definition

An expansion of the a trigonometric function for a compound angle in terms of trigonometric functions of same angles of the compound angle is called compound angle identity.

Identities

Sine of Compound angle

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

$(2) \,\,\,\,$ $\sin(A-B) = \sin A \cos B -\cos A \sin B$

Cosine of Compound angle

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

$(2) \,\,\,\,$ $\cos(A-B) = \cos A \cos B + \sin A \sin B$

Tangent of Compound angle

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

$(2) \,\,\,\,$ $\tan (A-B) = \dfrac{\tan A -\tan B}{1+\tan A \tan B}$

Cotangent of Compound angle

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

$(2) \,\,\,\,$ $\cot (A-B) = \dfrac{\cot B \cot A +1}{\cot B -\cot A}$

Additional Identities

$(1) \,\,\,\,$ $\sin(A+B) \sin(A-B) = \sin^2 A -\sin^2 B \,$ (or) $\, \cos^2 B -\cos^2 A$

$(2) \,\,\,\,$ $\cos(A+B) \cos(A-B) = \cos^2 A -\sin^2 B \,$ (or) $\, \cos^2 B -\sin^2 A$

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