## Trigonometric Identities

 Pythagorean Identities $\begin{cases}{\cos }^{2}t+{\sin }^{2}t=1\\ 1+{\tan }^{2}t={\sec }^{2}t\\ 1+{\cot }^{2}t={\csc }^{2}t\end{cases}$ Even-Odd Identities $\begin{cases}\cos \left(-t\right)=\cos t\hfill \\ \sec \left(-t\right)=\sec t\hfill \\ \sin \left(-t\right)=-\sin t\hfill \\ \tan \left(-t\right)=-\tan t\hfill \\ \csc \left(-t\right)=-\csc t\hfill \\ \cot \left(-t\right)=-\cot t\hfill \end{cases}$ Cofunction Identities $\begin{cases}\cos t=\sin \left(\frac{\pi }{2}-t\right)\hfill \\ \sin t=\cos \left(\frac{\pi }{2}-t\right)\hfill \\ \tan t=\cot \left(\frac{\pi }{2}-t\right)\hfill \\ \cot t=\tan \left(\frac{\pi }{2}-t\right)\hfill \\ \sec t=\csc \left(\frac{\pi }{2}-t\right)\hfill \\ \csc t=\sec \left(\frac{\pi }{2}-t\right)\hfill \end{cases}$ Fundamental Identities $\begin{cases}\tan t=\frac{\sin t}{\cos t}\hfill \\ \sec t=\frac{1}{\cos t}\hfill \\ \csc t=\frac{1}{\sin t}\hfill \\ \text{cot}t=\frac{1}{\text{tan}t}=\frac{\text{cos}t}{\text{sin}t}\hfill \end{cases}$ Sum and Difference Identities $\begin{cases}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \hfill \\ \cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \hfill \\ \sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \hfill \\ \sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \hfill \\ \tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\hfill \\ \tan \left(\alpha -\beta \right)=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\hfill \end{cases}$ Double-Angle Formulas $\begin{cases}\sin \left(2\theta \right)=2\sin \theta \cos \theta \hfill \\ \cos \left(2\theta \right)={\cos }^{2}\theta -{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=1 - 2{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=2{\cos }^{2}\theta -1\hfill \\ \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }\hfill \end{cases}$ Half-Angle Formulas $\begin{cases}\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\hfill \\ \cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\hfill \\ \tan \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\hfill \\ \tan \frac{\alpha }{2}=\frac{\sin \alpha }{1+\cos \alpha }\hfill \\ \tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }\hfill \end{cases}$ Reduction Formulas $\begin{cases}{\sin }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{2}\\ {\cos }^{2}\theta =\frac{1+\cos \left(2\theta \right)}{2}\\ {\tan }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{1+\cos \left(2\theta \right)}\end{cases}$ Product-to-Sum Formulas $\begin{cases}\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right]\hfill \\ \sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]\hfill \\ \sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right]\hfill \\ \cos \alpha \sin \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)-\sin \left(\alpha -\beta \right)\right]\hfill \end{cases}$ Sum-to-Product Formulas $\begin{cases}\sin \alpha +\sin \beta =2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \sin \alpha -\sin \beta =2\sin \left(\frac{\alpha -\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)\hfill \\ \cos \alpha -\cos \beta =-2\sin \left(\frac{\alpha +\beta }{2}\right)\sin \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \cos \alpha +\cos \beta =2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \end{cases}$ Law of Sines $\begin{cases}\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}\hfill \\ \frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }\hfill \end{cases}$ Law of Cosines $\begin{cases}{a}^{2}={b}^{2}+{c}^{2}-2bc\cos \alpha \hfill \\ {b}^{2}={a}^{2}+{c}^{2}-2ac\cos \beta \hfill \\ {c}^{2}={a}^{2}+{b}^{2}-2ab\text{cos}\gamma \hfill \end{cases}$