Sum Difference Identities Calculator

Sum and Difference Trigonometric Identities

Sum and difference identities rewrite trigonometric functions of combined angles as expressions involving the separate angles $\alpha$ and $\beta$. They are essential when an angle is not a standard value but can be split into familiar parts, such as $75^\circ = 30^\circ + 45^\circ$.

Sine and Cosine Identities

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$

$$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$$

$$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$

$$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$$

Tangent and Other Functions

$$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$$

Similar identities exist for cotangent, secant, and cosecant. Setting $\alpha = \beta$ yields the double-angle formulas.

See also: Double Angle Formula Calculator, Half Angle Calculator, and Exact Value of Trig Functions Calculator.