trigonometric identity

The so-called trigonometric identities are a useful set of equations that often allow one to make substitutions in an expression containing trigonometric functions, in order to simplify the expression or to put it in a more useful form. Most trig identities are actually quite easy to derive algebraically from their definitions, and every student of mathematics should derive them all at least once. Thereafter, it will no longer be necessary to memorize them; you will be able to derive them whenever they are needed.

We demonstrate one identity here—a form of the Pythagorean identity—to help you get started.

\[ \begin{eqnarray*} \sin^2 \varphi + \cos^2 \varphi& = & 1 \\ & & \\ & & \\ \left(\frac{\mbox{opp}}{\mbox{rad}}\right)^2 + \left(\frac{\mbox{adj}}{\mbox{rad}}\right)^2 & = & 1 \\ & & \\ & & \\ \frac{\mbox{opp}^2}{\mbox{rad}^2} + \frac{\mbox{adj}^2}{\mbox{rad}^2} & = & 1 \\ & & \\ & & \\ \mbox{opp}^2 + \mbox{adj}^2 & = & \mbox{rad}^2 \\ \end{eqnarray*} \]

…and the last equation follows from the Pythagorean Theorem. The proofs of the other identities listed below are similar. These identities are also available, together with other valuable trig stuff, on a downloadable Trig Reference Sheet.

Pythagorean Identities

\[\begin{eqnarray*} \sin^2 \varphi + \cos^2 \varphi& = & 1 \\ & & \\ & & \\ 1 + \tan^2 \varphi & = & \sec^2 \varphi \\ & & \\ & & \\ 1 + \cot^2 \varphi & = & \csc^2 \varphi \\ \end{eqnarray*} \]

Cofunction Identities

\[ \begin{eqnarray*} \sin\left(\frac{\pi}{2} - \varphi\right) & = & \cos \varphi \\ & & \\ & & \\ \csc\left(\frac{\pi}{2} - \varphi\right) & = & \sec \varphi \\ & & \\ & & \\ \sec\left(\frac{\pi}{2} - \varphi\right) & = & \csc \varphi \\ & & \\ & & \\ \cos\left(\frac{\pi}{2} - \varphi\right) & = & \sin \varphi \\ & & \\ & & \\ \tan\left(\frac{\pi}{2} - \varphi\right) & = & \cot \varphi \\ & & \\ & & \\ \cot\left(\frac{\pi}{2} - \varphi\right) & = & \tan \varphi \\ \end{eqnarray*} \]

Reduction Formulas

\[ \begin{eqnarray*} \sin( - \varphi ) & = & -\sin \varphi \\ & & \\ & & \\ \csc( - \varphi ) & = & -\csc \varphi \\ & & \\ & & \\ \sec( - \varphi ) & = & \sec \varphi \\ & & \\ & & \\ \cos( - \varphi ) & = & \cos \varphi \\ & & \\ & & \\ \tan( - \varphi ) & = & -\tan \varphi \\ & & \\ & & \\ \cot( - \varphi ) & = & -\cot \varphi \\ \end{eqnarray*} \]

Sum and Difference Formulas

\[ \begin{eqnarray*} \sin( \varphi \pm \theta ) & = & \sin \varphi \cos \theta \pm \cos \varphi \sin \theta \\ & & \\ & & \\ \cos( \varphi \pm \theta ) & = & \cos \varphi \cos \theta \mp \sin \varphi \sin \theta \\ & & \\ & & \\ \tan( \varphi \pm \theta ) & = & \frac{\tan \varphi \pm \tan \theta}{1 \mp \tan \varphi \tan \theta} \\ \end{eqnarray*} \]

Double-Angle Formulas

\[ \begin{eqnarray*} \sin 2\varphi & = & 2 \sin \varphi \cos \varphi \\ & & \\ & & \\ \cos 2\varphi & = & \cos^2 \varphi - \sin^2 \varphi \\ & & \\ & & \\ & = & 2 \cos^2 \varphi - 1 \\ & & \\ & & \\ & = & 1 - 2 \sin^2 \varphi \\ & & \\ & & \\ \tan 2\varphi & = & \frac{2 \tan \varphi}{1 - \tan^2 \varphi} \\ \end{eqnarray*} \]

Power-Reducing Formulas

\[ \begin{eqnarray*} \sin^2 \varphi & = & \frac{1 - \cos 2 \varphi}{2} \\ & & \\ & & \\ \cos^2 \varphi & = & \frac{1 + \cos 2 \varphi}{2} \\ & & \\ & & \\ \tan^2 \varphi & = & \frac{1 - \cos 2 \varphi}{1 + \cos 2 \varphi} \\ \end{eqnarray*} \]

Sum-to-Product Formulas

\[ \begin{eqnarray*} \sin \varphi + \sin \theta & = & 2 \sin \left(\frac{\varphi + \theta}{2}\right) \cos \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \sin \varphi - \sin \theta & = & 2 \cos \left(\frac{\varphi + \theta}{2}\right) \sin \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \cos \varphi + \cos \theta & = & 2 \cos \left(\frac{\varphi + \theta}{2}\right) \cos \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \cos \varphi - \cos \theta & = & -2 \sin \left(\frac{\varphi + \theta}{2}\right) \sin \left(\frac{\varphi - \theta}{2}\right) \\ \end{eqnarray*} \]

Product-to-Sum Formulas

\[ \begin{eqnarray*} \sin \varphi \sin \theta & = & \frac{1}{2}\left( \cos (\varphi - \theta) - \cos (\theta + \varphi) \right) \\ & & \\ & & \\ \cos \varphi \cos \theta & = & \frac{1}{2}\left( \cos (\varphi - \theta) + \cos (\theta + \varphi) \right) \\ & & \\ & & \\ \sin \varphi \cos \theta & = & \frac{1}{2}\left( \sin (\varphi + \theta) + \sin (\theta - \varphi) \right) \\ & & \\ & & \\ \cos \varphi \sin \theta & = & \frac{1}{2}\left( \sin (\varphi + \theta) - \sin (\theta - \varphi) \right) \\ \end{eqnarray*} \]