Trigonometric identities
Reciprocal identities
\begin{align}
\csc x &= \dfrac{1}{\sin x} \\
\sec x &= \dfrac{1}{\cos x} \\
\cot x &= \dfrac{1}{\tan x}
\end{align}
Quotient identities
\begin{align}
\tan x &= \dfrac{\sin x}{\cos x} \\
\cot x &= \dfrac{\cos x}{\sin x}
\end{align}
Periodic identities
The behaviour of any periodic function repeats itself every one full cycle \((2\pi)\), which means that adding or subtracting any whole number of cycles to an angle gives the exact same trigonometric ratio. Therefore, for any \(n\in\mathbb{Z}\),
\begin{align}
\sin (x+2\pi n) & = \sin x \\
\cos (x+2\pi n) & = \cos x \\
\tan (x+2\pi n) & = \tan x \\
\cot (x+2\pi n) & = \cot x \\
\sec (x+2\pi n) & = \sec x \\
\csc (x+2\pi n) & = \csc x
\end{align}
Symmetry identities
\begin{align}
\sin(-x) & = -\sin x; & \csc(-x) & = -\csc x \\
\cos(-x) & = \phantom{-}\cos x; & \sec(-x) & = \phantom{-}\sec x \\
\tan(-x) & = -\tan x; & \cot(-x) & = -\cot x \\
\end{align}
Co-ratio identities
The following are co-ratios of one another. \[\begin{aligned} \text{sine } (\sin) & \leftrightarrow \color{red}{\text{co}}\text{sine } (\cos) \\ \text{tangent } (\tan) & \leftrightarrow \color{red}{\text{co}}\text{tangent } (\cot) \\ \text{secant } (\sec) & \leftrightarrow \color{red}{\text{co}}\text{secant } (\csc) \end{aligned}\]
The general rule is as follows, and is easily applicable to quadrant I. \[\operatorname{ratio}(\text{angle})=\pm\operatorname{co-ratio}(\text{complementary angle})\] The rule can be extended to quadrants other than the first by using reference-angle identities in conjunction.
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\(\begin{aligned}
\sin\left(\tfrac{\pi}{2}+x\right) &= \phantom{-}\cos x \\
\cos\left(\tfrac{\pi}{2}+x\right) &= -\sin x \\
\tan\left(\tfrac{\pi}{2}+x\right) &= -\cot x \\
\cot\left(\tfrac{\pi}{2}+x\right) &= -\tan x \\
\sec\left(\tfrac{\pi}{2}+x\right) &= -\csc x \\
\csc\left(\tfrac{\pi}{2}+x\right) &= \phantom{-}\sec x \\
\end{aligned}\)
|
\(\begin{aligned}
\sin\left(\tfrac{\pi}{2}-x\right) &= \cos x \\
\cos\left(\tfrac{\pi}{2}-x\right) &= \sin x \\
\tan\left(\tfrac{\pi}{2}-x\right) &= \cot x \\
\cot\left(\tfrac{\pi}{2}-x\right) &= \tan x \\
\sec\left(\tfrac{\pi}{2}-x\right) &= \csc x \\
\csc\left(\tfrac{\pi}{2}-x\right) &= \sec x \\
\end{aligned}\)
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|
\(\begin{aligned}
\sin\left(\tfrac{3\pi}{2}-x\right) &= -\cos x \\
\cos\left(\tfrac{3\pi}{2}-x\right) &= -\sin x \\
\tan\left(\tfrac{3\pi}{2}-x\right) &= \phantom{-}\cot x \\
\cot\left(\tfrac{3\pi}{2}-x\right) &= \phantom{-}\tan x \\
\sec\left(\tfrac{3\pi}{2}-x\right) &= -\csc x \\
\csc\left(\tfrac{3\pi}{2}-x\right) &= -\sec x \\
\end{aligned}\)
|
\(\begin{aligned}
\sin\left(\tfrac{3\pi}{2}+x\right) &= -\cos x \\
\cos\left(\tfrac{3\pi}{2}+x\right) &= \phantom{-}\sin x \\
\tan\left(\tfrac{3\pi}{2}+x\right) &= -\cot x \\
\cot\left(\tfrac{3\pi}{2}+x\right) &= -\tan x \\
\sec\left(\tfrac{3\pi}{2}+x\right) &= \phantom{-}\csc x \\
\csc\left(\tfrac{3\pi}{2}+x\right) &= -\sec x \\
\end{aligned}\)
|
\(\begin{aligned}
\sin\left(\tfrac{\pi}{2}-x\right) &= \cos x \\
\cos\left(\tfrac{\pi}{2}-x\right) &= \sin x \\
\tan\left(\tfrac{\pi}{2}-x\right) &= \cot x \\
\cot\left(\tfrac{\pi}{2}-x\right) &= \tan x \\
\sec\left(\tfrac{\pi}{2}-x\right) &= \csc x \\
\csc\left(\tfrac{\pi}{2}-x\right) &= \sec x \\
\end{aligned}\)
\(\begin{aligned}
\sin\left(\tfrac{\pi}{2}+x\right) &= \phantom{-}\cos x \\
\cos\left(\tfrac{\pi}{2}+x\right) &= -\sin x \\
\tan\left(\tfrac{\pi}{2}+x\right) &= -\cot x \\
\cot\left(\tfrac{\pi}{2}+x\right) &= -\tan x \\
\sec\left(\tfrac{\pi}{2}+x\right) &= -\csc x \\
\csc\left(\tfrac{\pi}{2}+x\right) &= \phantom{-}\sec x \\
\end{aligned}\)
\(\begin{aligned}
\sin\left(\tfrac{3\pi}{2}-x\right) &= -\cos x \\
\cos\left(\tfrac{3\pi}{2}-x\right) &= -\sin x \\
\tan\left(\tfrac{3\pi}{2}-x\right) &= \phantom{-}\cot x \\
\cot\left(\tfrac{3\pi}{2}-x\right) &= \phantom{-}\tan x \\
\sec\left(\tfrac{3\pi}{2}-x\right) &= -\csc x \\
\csc\left(\tfrac{3\pi}{2}-x\right) &= -\sec x \\
\end{aligned}\)
\(\begin{aligned}
\sin\left(\tfrac{3\pi}{2}+x\right) &= -\cos x \\
\cos\left(\tfrac{3\pi}{2}+x\right) &= \phantom{-}\sin x \\
\tan\left(\tfrac{3\pi}{2}+x\right) &= -\cot x \\
\cot\left(\tfrac{3\pi}{2}+x\right) &= -\tan x \\
\sec\left(\tfrac{3\pi}{2}+x\right) &= \phantom{-}\csc x \\
\csc\left(\tfrac{3\pi}{2}+x\right) &= -\sec x \\
\end{aligned}\)
Reference-angle identities
The general rule is as follows. \[\operatorname{ratio}(\text{angle})=\pm\operatorname{ratio}(\text{reference angle})\] Note that the ratio remains the same. The sign \(\pm\) of the ratio depends on the quadrant of the angle. In the first quadrant, the angle is its own reference angle, and the relation is trivial.
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\(\begin{aligned}
\sin(\pi-x) &= \phantom{-}\sin x \\
\cos(\pi-x) &= -\cos x \\
\tan(\pi-x) &= -\tan x \\
\cot(\pi-x) &= -\cot x \\
\sec(\pi-x) &= -\sec x \\
\csc(\pi-x) &= \phantom{-}\csc x
\end{aligned}\)
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\(\begin{aligned}
\sin(\pi+x) &= -\sin x \\
\cos(\pi+x) &= -\cos x \\
\tan(\pi+x) &= \phantom{-}\tan x \\
\cot(\pi+x) &= \phantom{-}\cot x \\
\sec(\pi+x) &= -\sec x \\
\csc(\pi+x) &= -\csc x
\end{aligned}\)
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\(\begin{aligned}
\sin(2\pi-x) &= -\sin x \\
\cos(2\pi-x) &= \phantom{-}\cos x \\
\tan(2\pi-x) &= -\tan x \\
\cot(2\pi-x) &= -\cot x \\
\sec(2\pi-x) &= \phantom{-}\sec x \\
\csc(2\pi-x) &= -\csc x
\end{aligned}\)
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\(\begin{aligned}
\sin(\pi-x) &= \phantom{-}\sin x \\
\cos(\pi-x) &= -\cos x \\
\tan(\pi-x) &= -\tan x \\
\cot(\pi-x) &= -\cot x \\
\sec(\pi-x) &= -\sec x \\
\csc(\pi-x) &= \phantom{-}\csc x
\end{aligned}\)
\(\begin{aligned}
\sin(\pi+x) &= -\sin x \\
\cos(\pi+x) &= -\cos x \\
\tan(\pi+x) &= \phantom{-}\tan x \\
\cot(\pi+x) &= \phantom{-}\cot x \\
\sec(\pi+x) &= -\sec x \\
\csc(\pi+x) &= -\csc x
\end{aligned}\)
\(\begin{aligned}
\sin(2\pi-x) &= -\sin x \\
\cos(2\pi-x) &= \phantom{-}\cos x \\
\tan(2\pi-x) &= -\tan x \\
\cot(2\pi-x) &= -\cot x \\
\sec(2\pi-x) &= \phantom{-}\sec x \\
\csc(2\pi-x) &= -\csc x
\end{aligned}\)
Pythagorean identities
\begin{align}
\sin^2 x + \cos^2 x & = 1 \\
\tan^2 x + 1 & = \sec^2 x \\
1 + \cot^2 x & = \csc^2 x
\end{align}
Compound angle identities
\begin{align}
\sin(x\pm y) &= \sin x\cos y\pm \cos x\sin y \\
\cos(x\pm y) &= \cos x\cos y\pm \sin x\sin y \\
\tan(x\pm y) &= \dfrac{\tan x\pm \tan y}{1\mp \tan x\tan y}
\end{align}
Double-angle identities
\begin{align}
\sin 2x &= 2\sin x\cos x \\
\cos 2x &= \cos^2 x - \sin^2 x \\
&= 2\cos^2 x - 1 \\
&= 1-2\sin^2 x \\
\tan 2x &= \dfrac{2\tan x}{1-\tan^2 x}
\end{align}