Trigonometric Identities

Sine and Cosine of Sum

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

$\sin \left({a + b}\right) = \sin a \cos b + \cos a \sin b$

where $\sin$ and $\cos$ are sine and cosine.

Corollary

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

$\sin \left({a - b}\right) = \sin a \cos b - \cos a \sin b$

Tangent of Sum

$\displaystyle \tan \left({a + b}\right) = \frac {\tan a + \tan b} {1 - \tan a \tan b}$

where $\tan$ is tangent.

Corollary

$\displaystyle \tan \left({a - b}\right) = \frac {\tan a - \tan b} {1 + \tan a \tan b}$

Sum of Squares of Sine and Cosine

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

where $\sin$ and $\cos$ are sine and cosine.

Corollaries

	\(\displaystyle \)	\(\displaystyle 1 + \tan^2 x\)	\(=\)	\(\displaystyle \sec^2 x\)	\(\displaystyle \)	(when $\cos x \ne 0$)
	\(\displaystyle \)	\(\displaystyle 1 + \cot^2 x\)	\(=\)	\(\displaystyle \csc^2 x\)	\(\displaystyle \)	(when $\sin x \ne 0$)

where:

• $\tan$ and $\cot$ are tangent and cotangent.
• $\sec$ and $\csc$ are secant and cosecant.

Double Angle Formulas for Sine and Cosine

• $\sin \left({2 \theta}\right) = 2 \sin \theta \cos \theta$
• $\cos \left({2 \theta}\right) = \cos^2 \theta - \sin^2 \theta$
• $\displaystyle \tan \left({2 \theta}\right) = \frac {2\tan \theta} {1 - \tan^2 \theta}$

where $\sin, \cos, \tan$ are sine, cosine and tangent.

Corollary

• $\cos \left({2 \theta}\right) = 2 \ \cos^2 \theta - 1$
• $\cos \left({2 \theta}\right) = 1 - 2 \ \sin^2 \theta$

Half Angle Formulas for Sine and Cosine

$\displaystyle (1): \quad \sin \frac \theta 2 = \pm \sqrt {\frac {1 - \cos \theta} {2}}$

$\displaystyle (2): \quad \cos \frac \theta 2 = \pm \sqrt {\frac {1 + \cos \theta} {2}}$

$\displaystyle (3): \quad \tan \frac \theta 2 = \frac {\sin \theta} {1 + \cos \theta} = \frac {1 - \cos \theta} {\sin \theta}$

Product-to-Sum Formulas

\((1):\)	\(\displaystyle \)	\(\displaystyle \cos \alpha \cos \beta\)	\(=\)	\(\displaystyle \frac {\cos \left({\alpha - \beta}\right) + \cos \left({\alpha + \beta}\right)} 2\)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \sin \alpha \sin \beta\)	\(=\)	\(\displaystyle \frac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2\)	\(\displaystyle \)
\((3):\)	\(\displaystyle \)	\(\displaystyle \sin \alpha \cos \beta\)	\(=\)	\(\displaystyle \frac {\sin \left({\alpha + \beta}\right) + \sin \left({\alpha - \beta}\right)} 2\)	\(\displaystyle \)
\((4):\)	\(\displaystyle \)	\(\displaystyle \cos \alpha \sin \beta\)	\(=\)	\(\displaystyle \frac {\sin \left({\alpha + \beta}\right) - \sin \left({\alpha - \beta}\right)} 2\)	\(\displaystyle \)

Sum-to-Product Formulas

\((1):\)	\(\displaystyle \)	\(\displaystyle \sin \alpha + \sin \beta\)	\(=\)	\(\displaystyle 2 \sin \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \sin \alpha - \sin \beta\)	\(=\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)
\((3):\)	\(\displaystyle \)	\(\displaystyle \cos \alpha + \cos \beta\)	\(=\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)
\((4):\)	\(\displaystyle \)	\(\displaystyle \cos \alpha - \cos \beta\)	\(=\)	\(\displaystyle -2 \sin \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)

Power Reduction Formulas

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

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

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

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

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

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

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

Sum of Reciprocals of One Plus and Minus Sine

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

Difference of Reciprocals of One Plus and Minus Sine

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

Sum of Secant and Tangent

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

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

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

Reciprocal of One Plus Cosecant

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