Key Equations

Pythagorean identities	$\begin{array}{l} \cos^{2} θ + \sin^{2} θ = 1 \\ 1 + \cot^{2} θ = \csc^{2} θ \\ 1 + \tan^{2} θ = \sec^{2} θ \end{array} \begin{array}{l} \cos^{2} θ + \sin^{2} θ = 1 \\ 1 + \cot^{2} θ = \csc^{2} θ \\ 1 + \tan^{2} θ = \sec^{2} θ \end{array}$
Even-odd identities	$\begin{matrix} \tan (- θ) & = & - \tan θ \\ \cot (- θ) & = & - \cot θ \\ \sin (- θ) & = & - \sin θ \\ \csc (- θ) & = & - \csc θ \\ \cos (- θ) & = & \cos θ \\ \sec (- θ) & = & \sec θ \end{matrix} \begin{matrix} \tan (- θ) & = & - \tan \\ \cot (- θ) & = & - \cot \\ \sin (- θ) & = & - \sin \\ \csc (- θ) & = & - \csc \\ \cos (- θ) & = & \cos \\ \sec (- θ) & = & \sec \end{matrix}$
Reciprocal identities	$\begin{matrix} \sin θ & = & \frac{1}{\csc θ} \\ \cos θ & = & \frac{1}{\sec θ} \\ \tan θ & = & \frac{1}{\cot θ} \\ \csc θ & = & \frac{1}{\sin θ} \\ \sec θ & = & \frac{1}{\cos θ} \\ \cot θ & = & \frac{1}{\tan θ} \end{matrix} \begin{matrix} \sin & = & \frac{1}{\csc} \\ \cos & = & \frac{1}{\sec} \\ \tan & = & \frac{1}{\cot} \\ \csc & = & \frac{1}{\sin} \\ \sec & = & \frac{1}{\cos} \\ \cot & = & \frac{1}{\tan} \end{matrix}$
Quotient identities	$\begin{matrix} \tan θ & = & \frac{\sin θ}{\cos θ} \\ \cot θ & = & \frac{\cos θ}{\sin θ} \end{matrix} \begin{matrix} \tan & = & \frac{\sin}{\cos} \\ \cot & = & \frac{\cos}{\sin} \end{matrix}$

Sum Formula for Cosine	$\cos (α + β) = \cos α \cos β - \sin α \sin β \cos (α + β) = \cos$
Difference Formula for Cosine	$\cos (α - β) = \cos α \cos β + \sin α \sin β \cos (α - β) = \cos$
Sum Formula for Sine	$\sin (α + β) = \sin α \cos β + \cos α \sin β \sin (α + β) = \sin$
Difference Formula for Sine	$\sin (α - β) = \sin α \cos β - \cos α \sin β \sin (α - β) = \sin$
Sum Formula for Tangent	$\tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} \tan (α + β) = \frac{\tan}{1 - \tan}$
Difference Formula for Tangent	$\tan (α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β} \tan (α - β) = \frac{\tan}{1 + \tan}$
Cofunction identities	$\begin{matrix} \sin θ & = & \cos (\frac{π}{2} - θ) \\ \cos θ & = & \sin (\frac{π}{2} - θ) \\ \tan θ & = & \cot (\frac{π}{2} - θ) \\ \cot θ & = & \tan (\frac{π}{2} - θ) \\ \sec θ & = & \csc (\frac{π}{2} - θ) \\ \csc θ & = & \sec (\frac{π}{2} - θ) \end{matrix} \begin{matrix} \sin & = & \cos (\frac{π}{2} - θ) \\ \cos & = & \sin (\frac{π}{2} - θ) \\ \tan & = & \cot (\frac{π}{2} - θ) \\ \cot & = & \tan (\frac{π}{2} - θ) \\ \sec & = & \csc (\frac{π}{2} - θ) \\ \csc & = & \sec (\frac{π}{2} - θ) \end{matrix}$

Double-angle formulas	$\begin{matrix} \sin (2 θ) & = & 2 \sin θ \cos θ \\ \cos (2 θ) & = & \cos^{2} θ - \sin^{2} θ \\ = & 1 - 2 \sin^{2} θ \\ = & 2 \cos^{2} θ - 1 \\ \tan (2 θ) & = & \frac{2 \tan θ}{1 - \tan^{2} θ} \end{matrix} \begin{matrix} \sin (2 θ) & = & 2 \sin \\ \cos (2 θ) & = & \cos^{2} θ - \sin^{2} θ \\ = 1 - 2 \sin^{2} θ \\ = 2 \cos^{2} θ - 1 \\ \tan (2 θ) & = & \frac{2 \tan}{1 - \tan^{2} θ} \end{matrix}$
Reduction formulas	$\begin{matrix} \sin^{2} θ & = & \frac{1 - \cos (2 θ)}{2} \\ \cos^{2} θ & = & \frac{1 + \cos (2 θ)}{2} \\ \tan^{2} θ & = & \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{matrix} \begin{matrix} \sin^{2} θ & = & \frac{1 - \cos (2 θ)}{2} \\ \cos^{2} θ & = & \frac{1 + \cos (2 θ)}{2} \\ \tan^{2} θ & = & \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{matrix}$
Half-angle formulas	$\begin{matrix} \sin \frac{α}{2} & = & \pm \sqrt{\frac{1 - \cos α}{2}} \\ \cos \frac{α}{2} & = & \pm \sqrt{\frac{1 + \cos α}{2}} \\ \tan \frac{α}{2} & = & \pm \sqrt{\frac{1 - \cos α}{1 + \cos α}} \\ = & \frac{\sin α}{1 + \cos α} \\ = & \frac{1 - \cos α}{\sin α} \end{matrix} \begin{matrix} \sin & = & \pm \sqrt{\frac{1 - \cos}{2}} \\ \cos & = & \pm \sqrt{\frac{1 + \cos}{2}} \\ \tan & = & \pm \sqrt{\frac{1 - \cos}{1 + \cos}} \\ = \frac{\sin}{1 + \cos} \\ = \frac{1 - \cos}{\sin} \end{matrix}$

Product-to-sum Formulas

\begin{matrix} \cos α \cos β & = & \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin α \cos β & = & \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin α \sin β & = & \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos α \sin β & = & \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{matrix} \begin{matrix} \cos & = & \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin & = & \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin & = & \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos & = & \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{matrix}

Sum-to-product Formulas

\begin{matrix} \sin α + \sin β & = & 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \\ \sin α - \sin β & = & 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2}) \\ \cos α - \cos β & = & - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2}) \\ \cos α + \cos β & = & 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \end{matrix} \begin{matrix} \sin & = & 2 \\ \sin & = & 2 \\ \cos & = & - 2 \\ \cos & = & 2 \end{matrix}