Trigonometric Identities

 Pythagorean Identities

\begin{array}{l} {\sin ^2}x + {\cos ^2}x = 1\\ {\sec ^2}x - {\tan ^2}x = 1\\ {\csc ^2}x - {\cot ^2}x = 1 \end{array}

Half Angle Formula

\begin{array}{l} {\sin ^2}x = \frac{1}{2}\left( {1 - \cos 2x\;} \right)\\ {\cos ^2}x = \frac{1}{2}\left( {1 + \cos 2x\;} \right)\\ {\tan ^2}x = \;\frac{{1 - \cos 2x}}{{1 + \cos 2x}}\\ \sin \frac{x}{2} = \pm \sqrt {\frac{{1 - \cos x}}{2}} \\ \cos \frac{x}{2} = \pm \sqrt {\frac{{1 + \cos x}}{2}} \\ \tan \frac{x}{2} = \pm \sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} \end{array}

Double Angle Formula

\begin{array}{l} \sin 2ax = 2\sin ax\cos ax\\ \cos 2ax = {\cos ^2}ax - {\sin ^2}ax\\ \cos 2ax = 1 - {\sin ^2}ax\\ \cos 2ax = 2\;{\cos ^2}ax - 1\\ \tan 2ax\; = \frac{{2\tan ax}}{{1 - {{\tan }^2}ax}} \end{array}

Sum To Product Formula

\begin{array}{l} \sin x + \sin y = 2\sin \left( {\frac{{x + y}}{2}} \right)\cos \left( {\frac{{x - y}}{2}} \right)\\ \sin x - \sin y = 2\cos \left( {\frac{{x + y}}{2}} \right)\sin \left( {\frac{{x - y}}{2}} \right)\\ \cos x + \cos y = 2\cos \left( {\frac{{x + y}}{2}} \right)\cos \left( {\frac{{x - y}}{2}} \right)\\ \cos x - \cos y = 2\sin \left( {\frac{{x + y}}{2}} \right)\sin \left( {\frac{{x - y}}{2}} \right) \end{array}

Product To Sum Formula

\begin{array}{l} \sin x\sin y = \frac{1}{2}(\cos \left( {x - y} \right) - \cos \left( {x + y} \right)\;)\\ \cos x\cos y = \frac{1}{2}\left( {\cos \left( {x - y} \right) + \cos \left( {x + y} \right)} \right)\\ \sin x\cos y = \frac{1}{2}\left( {\sin \left( {x - y} \right) + \sin \left( {x + y} \right)} \right)\\ \cos x\sin y = \frac{1}{2}\left( {\sin \left( {x + y} \right) - \sin \left( {x - y} \right)} \right) \end{array}

Sum And Difference Formula

\begin{array}{l} \sin \left( {x + y} \right) = \sin x\cos x + \cos x\sin x\\ \sin \left( {x - y} \right) = \sin x\cos x - \cos x\sin x\\ \cos \left( {x + y} \right) = \cos x\cos y - \sin x\sin y\\ \cos \left( {x - y} \right) = \cos x\cos y + \sin x\sin y\\ \tan \left( {x + y} \right) = \frac{{\tan x + \tan y}}{{1 - \tan x\tan y}}\\ \tan \left( {x - y} \right) = \frac{{\tan x - \tan y}}{{1 + \tan x\tan y}} \end{array}