Trig Formulas & Identities

This purely reference page lists some of the most important trig formulas and identities, including relationships among trig functions, addition and subtraction formulas, double-angle formulas, and half-angle formulas.

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Relationships Among Trig Functions

csc⁡𝑥=1sin⁡𝑥sec⁡𝑥=1cos⁡𝑥tan⁡𝑥=sin⁡𝑥cos⁡𝑥cot⁡𝑥=cos⁡𝑥sin⁡𝑥
sin2⁡𝑥+cos2⁡𝑥=11+cot2⁡𝑥=csc2⁡𝑥1+tan2⁡𝑥=sec2⁡𝑥

Negative Angles

sin⁡(−𝑥)=−sin⁡𝑥csc⁡(−𝑥)=−csc⁡𝑥cos⁡(−𝑥)=cos⁡𝑥sec⁡(−𝑥)=sec⁡𝑥tan⁡(−𝑥)=−tan⁡𝑥cot⁡(−𝑥)=−cot⁡𝑥

Addition and Subtraction Formulas

sin⁡(𝑥+𝑦)=sin⁡𝑥cos⁡𝑦+sin⁡𝑦cos⁡𝑥sin⁡(𝑥−𝑦)=sin⁡𝑥cos⁡𝑦−sin⁡𝑦cos⁡𝑥cos⁡(𝑥+𝑦)=cos⁡𝑥cos⁡𝑦−sin⁡𝑥sin⁡𝑦cos⁡(𝑥−𝑦)=cos⁡𝑥cos⁡𝑦+sin⁡𝑥sin⁡𝑦tan⁡(𝑥+𝑦)=tan⁡𝑥+tan⁡𝑦1−tan⁡𝑥tan⁡𝑦tan⁡(𝑥−𝑦)=tan⁡𝑥−tan⁡𝑦1+tan⁡𝑥tan⁡𝑦

Double-Angle Formulas

sin⁡2𝑥=2sin⁡𝑥cos⁡𝑥cos⁡2𝑥=cos2⁡𝑥−sin2⁡𝑥=1−2sin2⁡𝑥=2cos2⁡𝑥−1tan⁡2𝑥=2tan⁡𝑥1−tan2⁡𝑥

Half-Angle Formulas

Particularly useful for integrating sin2⁡𝑥 and cos2⁡𝑥: sin2⁡𝑥=12−12cos⁡2𝑥cos2⁡𝑥=12+12cos⁡2𝑥