Trigonometric Identities

Addition 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

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

Subtraction Formulas

Sine of Difference

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

Cosine of Difference

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

Tangent of Difference

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

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

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

Half Angle Formulas

Half Angle Formula for Sine

\(\ds \sin \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\frac {1 - \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$

\(\ds \sin \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\dfrac {1 - \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text {III}$ or quadrant $\text {IV}$

Half Angle Formula for Cosine

\(\ds \cos \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\frac {1 + \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$

\(\ds \cos \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\frac {1 + \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {III}$

Half Angle Formula for Tangent

\(\ds \tan \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {III}$

\(\ds \tan \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\)

for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {IV}$

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

When $\theta = \paren {2 k + 1} \pi$, $\tan \dfrac \theta 2$ 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

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

Sine by Cosine

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

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

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

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

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

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

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$

Sine Form

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \sin {x + \arctan \dfrac q p}$

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

$\tan x \paren {\tan x + \cot x} = \sec^2 x$

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

$\csc x + \cot x = \cot {\dfrac x 2}$

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

$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$

Reciprocal of One Minus Sine

$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$

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

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

Reciprocal of One Plus Cosine

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

Reciprocal of One Minus Cosine

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

Sum of Secant and Tangent

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

Cosine over Sum of Secant and Tangent

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

Secant Plus One over Secant Squared

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

Sine Plus Cosine times Tangent Plus Cotangent

$\paren {\sin x + \cos x} \paren {\tan x + \cot x} = \sec x + \csc x$

Tangent over Secant Plus One

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

Squares of Linear Combination of Sine and Cosine

$\paren {a \cos x + b \sin x}^2 + \paren {b \cos x - a \sin x}^2 = a^2 + b^2$

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

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