Trigonometric Identities

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Fundamental Identities Reciprocal Identities: $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$ Quotient Identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$ Pythagorean Identities: $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ Angle Sum and Difference Identities $\sin(A+B) = \sin A \cos B + \cos A \sin B$ $\sin(A-B) = \sin A \cos B - \cos A \sin B$ $\cos(A+B) = \cos A \cos B - \sin A \sin B$ $\cos(A-B) = \cos A \cos B + \sin A \sin B$ $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ $\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ Double Angle Identities $\sin(2\theta) = 2 \sin \theta \cos \theta$ $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$ $\cos(2\theta) = 2 \cos^2 \theta - 1$ $\cos(2\theta) = 1 - 2 \sin^2 \theta$ $\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}$ Half Angle Identities $\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$ $\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$ $\tan \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$ $\tan \left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta}$ $\tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}$ Product-to-Sum Identities $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$ $\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$ $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$ $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$ Sum-to-Product Identities $\sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$ $\sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ $\cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$ $\cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ Cofunction Identities $\sin \left(\frac{\pi}{2} - \theta\right) = \cos \theta$ $\cos \left(\frac{\pi}{2} - \theta\right) = \sin \theta$ $\tan \left(\frac{\pi}{2} - \theta\right) = \cot \theta$ $\csc \left(\frac{\pi}{2} - \theta\right) = \sec \theta$ $\sec \left(\frac{\pi}{2} - \theta\right) = \csc \theta$ $\cot \left(\frac{\pi}{2} - \theta\right) = \tan \theta$ Periodicity Identities $\sin(\theta + 2\pi k) = \sin \theta$ $\cos(\theta + 2\pi k) = \cos \theta$ $\tan(\theta + \pi k) = \tan \theta$ where $k$ is an integer. Even/Odd Identities $\sin(-\theta) = -\sin \theta$ (Odd) $\cos(-\theta) = \cos \theta$ (Even) $\tan(-\theta) = -\tan \theta$ (Odd) $\csc(-\theta) = -\csc \theta$ (Odd) $\sec(-\theta) = \sec \theta$ (Even) $\cot(-\theta) = -\cot \theta$ (Odd) Triple Angle Identities $\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$ $\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$ $\tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$ Power Reduction Formulas $\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$ $\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$ $\tan^2 \theta = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$