Trigonometric Identities

Reciprocal identities

Cosecant

\csc x = \frac{1}{\sin x}

Secant

\sec x = \frac{1}{\cos x}

Cotangent

\cot x = \frac{1}{\tan x}

Tangent

\tan x = \frac{\sin x}{\cos x}

cos x ≠ 0

Cotangent

\cot x = \frac{\cos x}{\sin x}

sin x ≠ 0

Sine and cosine

\sin^{2} x + \cos^{2} x = 1

Tangent and secant

1 + \tan^{2} x = \sec^{2} x

Cotangent and cosecant

1 + \cot^{2} x = \csc^{2} x

Sine is odd

\sin (- x) = - \sin x

Cosine is even

\cos (- x) = \cos x

Tangent is odd

\tan (- x) = - \tan x

Cosecant is odd

\csc (- x) = - \csc x

Secant is even

\sec (- x) = \sec x

Cotangent is odd

\cot (- x) = - \cot x

Sine

\sin (\frac{π}{2} - x) = \cos x

x in radians

Cosine

\cos (\frac{π}{2} - x) = \sin x

x in radians

Tangent

\tan (\frac{π}{2} - x) = \cot x

x in radians

Cotangent

\cot (\frac{π}{2} - x) = \tan x

x in radians

Secant

\sec (\frac{π}{2} - x) = \csc x

x in radians

Cosecant

\csc (\frac{π}{2} - x) = \sec x

x in radians

Sine, supplement

\sin (π - x) = \sin x

Cosine, supplement

\cos (π - x) = - \cos x

Tangent, supplement

\tan (π - x) = - \tan x

Sine, half-turn

\sin (π + x) = - \sin x

Cosine, half-turn

\cos (π + x) = - \cos x

Tangent has period π

\tan (π + x) = \tan x

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