Trigonometric Identities

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Contents

Compound Angle Formulas

Sine of Sum

$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$


Cosine of Sum

$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$


Tangent of Sum

$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$

where $\tan$ is tangent.


Sum of Squares of Sine and Cosine

$\cos^2 x + \sin^2 x = 1$


Corollary 1

$\sec^2 x - \tan^2 x = 1 \quad \text {(when $\cos x \ne 0$)}$


Corollary 2

$\csc^2 x - \cot^2 x = 1 \quad \text {(when $\sin x \ne 0$)}$


Double Angle Formulas

Double Angle Formula for Sine

$\sin 2 \theta = 2 \sin \theta \cos \theta$


Double Angle Formula for Cosine

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$


Double Angle Formula for Tangent

$\map \tan {2 \theta} = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$


Half Angle Formulas

Half Angle Formula for Sine

\(\displaystyle \sin \frac \theta 2\) \(=\) \(\displaystyle +\sqrt {\frac {1 - \cos \theta} 2}\) for $\dfrac \theta 2$ in quadrant I or quadrant II
\(\displaystyle \sin \frac \theta 2\) \(=\) \(\displaystyle -\sqrt {\dfrac {1 - \cos \theta} 2}\) for $\dfrac \theta 2$ in quadrant III or quadrant IV


Half Angle Formula for Cosine

\(\displaystyle \cos \frac \theta 2\) \(=\) \(\displaystyle + \sqrt {\frac {1 + \cos \theta} 2}\) for $\dfrac \theta 2$ in quadrant I or quadrant IV
\(\displaystyle \cos \frac \theta 2\) \(=\) \(\displaystyle - \sqrt {\frac {1 + \cos \theta} 2}\) for $\dfrac \theta 2$ in quadrant II or quadrant III


Half Angle Formula for Tangent

\(\displaystyle \tan \frac \theta 2\) \(=\) \(\displaystyle + \sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\) for $\dfrac \theta 2$ in quadrant I or quadrant III
\(\displaystyle \tan \frac \theta 2\) \(=\) \(\displaystyle - \sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\) for $\dfrac \theta 2$ in quadrant II or quadrant IV

where $\tan$ denotes tangent and $\cos$ denotes cosine.

When $\theta = \left({2 k + 1}\right) \pi$, $\tan \dfrac \theta 2$ is undefined.


Half Angle Formula for Tangent: Corollary 1

$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$


Half Angle Formula for Tangent: Corollary 2

$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$


Half Angle Formula for Tangent: Corollary 3

$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$


One Plus Tangent Half Angle over One Minus Tangent Half Angle

$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$


Simpson's Formulas

Cosine by Cosine

$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$


Sine by Sine

$\sin \alpha \sin \beta = \dfrac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2$


Sine by Cosine

$\sin \alpha \cos \beta = \dfrac {\sin \left({\alpha + \beta}\right) + \sin \left({\alpha - \beta}\right)} 2$


Cosine by Sine

$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$


Prosthaphaeresis Formulas

Sine plus Sine

$\sin \alpha + \sin \beta = 2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$


Sine minus Sine

$\sin \alpha - \sin \beta = 2 \cos \left({\dfrac {\alpha + \beta} 2}\right) \sin \left({\dfrac {\alpha - \beta} 2}\right)$


Cosine plus Cosine

$\cos \alpha + \cos \beta = 2 \, \map \cos {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$


Cosine minus Cosine

$\cos \alpha - \cos \beta = -2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \sin {\dfrac {\alpha - \beta} 2}$


Sum Formulas for Sine and Cosine

Sine of x plus Cosine of x: Sine Form

$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$


Sine of x plus Cosine of x: Cosine Form

$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$


Sine of x minus Cosine of x: Sine Form

$\sin x - \cos x = \sqrt 2 \sin \left({x - \dfrac \pi 4}\right)$


Sine of x minus Cosine of x: Cosine Form

$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$


Cosine of x minus Sine of x: Sine Form

$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$


Cosine of x minus Sine of x: Cosine Form

$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$


Multiple of Sine plus Multiple of Cosine

Cosine Form

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \cos \left({x + \arctan \dfrac {-p} q}\right)$


Sine Form

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \sin \left({x + \arctan \dfrac q p}\right)$


Power Reduction Formulas

Square of Sine

$\sin^2 x = \dfrac {1 - \cos 2 x} 2$


Square of Cosine

$\cos^2 x = \dfrac {1 + \cos 2 x} 2$


Square of Tangent

$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$


Cube of Sine

$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$


Cube of Cosine

$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$


Fourth Power of Sine

$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$


Fourth Power of Cosine

$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$


Fifth Power of Sine

$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$


Fifth Power of Cosine

$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$


Minor Identities

Sum of Tangent and Cotangent

$\tan x + \cot x = \sec x \csc x$


Tangent times Tangent plus Cotangent

$\tan x \left({\tan x + \cot x}\right) = \sec^2 x$


Secant Minus Cosine

$\sec x - \cos x = \sin x \tan x$


Square of Tangent Minus Square of Sine

$\tan^2 x - \sin^2 x = \tan^2 x \ \sin^2 x$


Difference of Fourth Powers of Cosine and Sine

$\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$


Cosecant Minus Sine

$\csc x - \sin x = \cos x \ \cot x$


Cotangent Minus Tangent

$\cot x - \tan x = 2 \cot 2 x$


Sum of Cosecant and Cotangent

$\displaystyle \csc x + \cot x = \cot {\frac x 2}$


Sum of Squares of Secant and Cosecant

$\sec^2 x + \csc^2 x = \sec^2 x \ \csc^2 x$


Difference of Fourth Powers of Secant and Tangent

$\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$


Reciprocal of One Plus Sine

$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \sec^2 \left({\dfrac \pi 4 - \dfrac x 2}\right)$


Reciprocal of One Minus Sine

$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \sec^2 \left({\dfrac \pi 4 + \dfrac x 2}\right)$


Sum of Reciprocals of One Plus and Minus Sine

$\dfrac 1 {1 - \sin x} + \dfrac 1 {1 + \sin x} = 2 \ \sec^2 x$


Difference of Reciprocals of One Plus and Minus Sine

$\displaystyle \frac 1 {1 - \sin x} - \frac 1 {1 + \sin x} = 2 \tan x \ \sec x$


Reciprocal of One Plus Cosine

$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$


Reciprocal of One Minus Cosine

$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \csc^2 \left({\dfrac x 2}\right)$


Sum of Secant and Tangent

$\sec x + \tan x = \dfrac {1 + \sin x} {\cos x}$


Cosine over Sum of Secant and Tangent

$\displaystyle \frac {\cos x} {\sec x + \tan x} = 1 - \sin x$


Secant Plus One over Secant Squared

$\displaystyle \frac {\sec x + 1} {\sec^2 x} = \frac {\sin^2 x} {\sec x - 1}$


Sine Plus Cosine times Tangent Plus Cotangent

$\left({\sin x + \cos x}\right) \left({\tan x + \cot x}\right) = \sec x + \csc x$


Tangent over Secant Plus One

$\displaystyle \frac {\tan x} {\sec x + 1} = \frac {\sec x - 1} {\tan x}$


Squares of Linear Combination of Sine and Cosine

$\left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2 = a^2 + b^2$


Reciprocal of One Minus Secant

$\dfrac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \dfrac 1 {1 - \sec x}$


Reciprocal of One Plus Cosecant

$\dfrac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \dfrac 1 {1 + \csc x}$