Trigonometric Formulas and Identities

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

$\cot \theta = \frac{\cos \theta}{\sin \theta}$

$\csc \theta = \frac{1}{\sin \theta}$

$\sec \theta = \frac{1}{\cos \theta}$

$\cot \theta = \frac{1}{\tan \theta}$

$\sin \theta = \frac{1}{\csc \theta}$

$\cos \theta = \frac{1}{\sec \theta}$

$\tan \theta = \frac{1}{\cot \theta}$

$\sin^2 \theta + \cos^2 \theta = 1$

$\tan^2 \theta + 1 = \sec^2 \theta$

$1 + \cot^2 \theta = \csc^2 \theta$

$\sin(-\theta) = -\sin \theta$

$\cos(-\theta) = \cos \theta$

$\tan(-\theta) = -\tan \theta$

$\csc(-\theta) = -\csc \theta$

$\sec(-\theta) = \sec \theta$

$\cot(-\theta) = -\cot \theta$

$\sin(\theta + 2\pi n) = \sin \theta$

$\cos(\theta + 2\pi n) = \cos \theta$

$\tan(\theta + \pi n) = \tan \theta$

$\csc(\theta + 2\pi n) = \csc \theta$

$\sec(\theta + 2\pi n) = \sec \theta$

$\cot(\theta + \pi n) = \cot \theta$

$\sin(2\theta) = 2 \sin \theta \cos \theta$

$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta $
$ \qquad = 2 \cos^2 \theta - 1 $
$ \qquad = 1 - 2 \sin^2 \theta$

$\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

$\sin^2 \theta = \frac{1}{2}(1 - \cos(2\theta))$

$\cos^2 \theta = \frac{1}{2}(1 + \cos(2\theta))$

$\tan^2 \theta = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$

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

$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

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

$\sin \alpha \sin \beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]$

$\cos \alpha \cos \beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]$

$\sin \alpha \cos \beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]$

$\sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)$

$\sin \alpha - \sin \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)$

$\cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)$

$\cos \alpha - \cos \beta = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)$

$\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$

$\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$

$\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta$

$\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta$

$\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta$

$\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta$

If $x$ is an angle in degrees and $t$ is an angle in radians, then: