## Trigonometric Identities & Formulas

 Pythagorean Identities $$\sin^2 \theta + \cos^2 \theta = 1$$ $$\tan^2 \theta + 1 = \sec^2 \theta$$ $$\cot^2 \theta + 1 = \csc^2 \theta$$
 Double Angle Formulas $$\sin (2 \theta) = 2 \sin \theta \cos \theta$$ $$\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta$$ $$\cos (2 \theta)$$ $$= 2 \cos^2 \theta - 1$$ $$\cos (2 \theta)$$ $$= 1 - 2 \sin^2 \theta$$ $$\tan (2 \theta) = {2 \tan \theta \over 1 - \tan^2 \theta }$$
 Half Angle Formulas $$\sin^2 \theta = \frac{1}{2} (1 - \cos (2 \theta))$$ $$\cos^2 \theta = \frac{1}{2} (1 + \cos (2 \theta))$$ $$\tan^2 \theta = {1 - \cos (2 \theta) \over 1 + \cos (2 \theta) }$$
 Sum & Difference Formulas $$\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$ $$\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$$ $$\tan (\alpha \pm \beta) = {\tan \alpha \pm \tan \beta \over 1 \mp \tan \alpha \tan \beta}$$
 Product to Sum Formulas $$\sin \alpha \sin \beta = \frac{1}{2} [\cos (\alpha - \beta) - \cos (\alpha + \beta)]$$ $$\cos \alpha \cos \beta = \frac{1}{2} [\cos (\alpha - \beta) + \cos (\alpha + \beta)]$$ $$\sin \alpha \cos \beta = \frac{1}{2} [\sin (\alpha + \beta) + \sin (\alpha - \beta)]$$ $$\cos \alpha \sin \beta = \frac{1}{2} [\sin (\alpha + \beta) - \sin (\alpha - \beta)]$$
 Sum to Product Formulas $$\sin \alpha + \sin \beta = 2 \sin ({\alpha + \beta \over 2}) \cos ({\alpha + \beta \over 2})$$ $$\sin \alpha - \sin \beta = 2 \cos ({\alpha + \beta \over 2}) \sin ({\alpha + \beta \over 2})$$ $$\cos \alpha + \cos \beta = 2 \cos ({\alpha + \beta \over 2}) \cos ({\alpha + \beta \over 2})$$ $$\cos \alpha - \cos \beta = -2 \sin ({\alpha + \beta \over 2}) \sin ({\alpha + \beta \over 2})$$

 $$\sin (\frac{\pi}{2} - \theta) = \cos \theta$$ $$\cos (\frac{\pi}{2} - \theta) = \sin \theta$$ $$\sec (\frac{\pi}{2} - \theta) = \csc \theta$$ $$\csc (\frac{\pi}{2} - \theta) = \sec \theta$$ $$\tan (\frac{\pi}{2} - \theta) = \cot \theta$$ $$\cot (\frac{\pi}{2} - \theta) = \tan \theta$$ $$\sin (\theta + 2 \pi n) = \sin \theta$$ $$\csc (\theta + 2 \pi n) = \csc \theta$$ $$\cos (\theta + 2 \pi n) = \cos \theta$$ $$\sec (\theta + 2 \pi n) = \sec \theta$$ $$\tan (\theta + \pi n) = \tan \theta$$ $$\cot (\theta + \pi n) = \cot \theta$$
 $$\sin (-\theta) = -\sin \theta$$ $$\csc (-\theta) = -\csc \theta$$ $$\cos (-\theta) = \cos \theta$$ $$\sec (-\theta) = \sec \theta$$ $$\tan (-\theta) = -\tan \theta$$ $$\cot (-\theta) = -\cot \theta$$

 $$\sin (\pi - \theta) = \sin \theta$$ $$\cos (\pi - \theta) = -\cos \theta$$
 $$\tan \theta = {\sin \theta \over \cos \theta}$$ $$\cot \theta = {\cos \theta \over \sin \theta}$$