PRIME

Platonic

Realms

Interactive

Mathematics

Encyclopedia

# 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

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

Contributors

• B. Sidney Smith, author

Citation Info

• [MLA] Smith, B. Sidney. "trigonometric identity." Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 3 Mar 2016. Web. 20 Nov 2018. <http://platonicrealms.com/>
• [APA] Smith, B. Sidney (3 Mar 2016). trigonometric identity. Retrieved 20 Nov 2018 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/trigonometric-identity/

Trigonometry Reference Sheet

Trigonometry facts and formulas, including definitions, identities, function values, and graphs. Specially formatted for printing and keeping in a notebook or ring-binder.

Trigonometry Word-Cloud

You can triangulate all the critical terms in the study of trigonometry with this beautiful, colorful word-cloud poster.

Trigonometry Wall Clock

It's easier to tell the time if you know all the angles.