Algebra and Precalculus A Practical Guide

\begin{align*} \boxed{ \begin{array}{c|l} \text{Reciprocal} \amp \begin{aligned} \amp \csc\theta = \dfrac{1}{\sin\theta}, \quad \sec\theta = \dfrac{1}{\cos\theta}, \quad \cot\theta = \dfrac{1}{\tan\theta} \\[6pt] \end{aligned} \\ \hline \text{Quotient} \amp \begin{aligned} \rule{0pt}{2em} \amp \tan\theta = \frac{\sin{\theta}}{\cos{\theta}} \quad \cot\theta = \frac{\cos\theta}{\sin{\theta}} \\[6pt] \end{aligned} \\ \hline \text{Pythagorean} \amp \begin{aligned} \rule{0pt}{1.5em} \amp \sin^{2}\theta + \cos^{2}\theta = 1 \\ \rule{0pt}{1.5em} \amp 1 + \tan^{2}\theta = \sec^{2}\theta \\ \rule{0pt}{1.5em} \amp 1 + \cot^{2}\theta = \csc^{2}\theta \\[6pt] \end{aligned} \\ \hline \text{Addition} \amp \begin{aligned} \rule{0pt}{1.5em} \sin(\alpha \pm \beta) \amp= \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \\ \rule{0pt}{1.5em} \cos(\alpha \pm \beta) \amp= \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \\ \rule{0pt}{2em} \tan(\alpha \pm \beta) \amp= \dfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta} \\[6pt] \end{aligned} \\ \hline \text{Double-angle} \amp \begin{aligned} \rule{0pt}{1.5em} \sin(2\theta) \amp= 2\sin\theta\cos\theta \\ \rule{0pt}{1.5em} \cos(2\theta) \amp= \cos^{2}\theta - \sin^{2}\theta \\ \rule{0pt}{1em} \amp= 2\cos^{2}\theta - 1 \\ \rule{0pt}{1em} \amp= 1 - 2\sin^{2}\theta \\ \rule{0pt}{2em} \tan(2\theta) \amp= \dfrac{2\tan\theta}{1 - \tan^{2}\theta} \end{aligned} \end{array}} \end{align*}