# Calculus/Table of Trigonometry

## Definitions

• $\tan(x)= \frac{\sin x}{\cos x}$
• $\sec(x)= \frac{1}{\cos x}$
• $\cot(x)= \frac{\cos x}{\sin x}= \frac{1}{\tan x}$
• $\csc(x)= \frac{1}{\sin x}$

## Pythagorean Identities

• $\sin^2 x + \cos^2 x =1 \$
• $1+\tan^2(x)= \sec^2 x \$
• $1+\cot^2(x)= \csc^2 x \$

## Double Angle Identities

• $\sin(2 x)= 2\sin x \cos x \$
• $\cos(2 x)= \cos^2 x - \sin^2 x \$
• $\tan(2x) = \frac{2 \tan (x)} {1 - \tan^2(x)}$
• $\cos^2(x) = {1 + \cos(2x) \over 2}$
• $\sin^2(x) = {1 - \cos(2x) \over 2}$

## Angle Sum Identities

$\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y$
$\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y$
$\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y$
$\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y$
$\sin x+\sin y=2\sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right)$
$\sin x-\sin y=2\cos \left( \frac{x+y}{2} \right) \sin \left( \frac{x-y}{2} \right)$
$\cos x+\cos y=2\cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right)$
$\cos x-\cos y=-2\sin \left( \frac{x+y}{2} \right)\sin \left( \frac{x-y}{2} \right)$
$\tan x+\tan y=\frac{\sin \left( x+y\right) }{\cos x\cos y}$
$\tan x-\tan y=\frac{\sin \left( x-y\right) }{\cos x\cos y}$
$\cot x+\cot y=\frac{\sin \left( x+y\right) }{\sin x\sin y}$
$\cot x-\cot y=\frac{-\sin \left( x-y\right) }{\sin x\sin y}$

## Product-to-sum identities

$\cos\left (x\right ) \cos\left (y\right ) = {\cos\left (x + y\right ) + \cos\left (x - y\right ) \over 2} \;$
$\sin\left (x\right ) \sin\left (y\right ) = {\cos\left (x - y\right ) - \cos\left (x + y\right ) \over 2} \;$
$\sin\left (x\right ) \cos\left (y\right ) = {\sin\left (x + y\right ) + \sin\left (x - y\right ) \over 2} \;$
$\cos\left (x\right ) \sin\left (y\right ) = {\sin\left (x + y\right ) - \sin\left (x - y\right ) \over 2} \;$