Trigonometric Identities

Quotient

 * $$\tan{\theta}\,=\frac{\sin{\theta}\,}{\cos{\theta}\,}\,$$


 * $$\cot{\theta}\,=\frac{\cos{\theta}\,}{\sin{\theta}\,}\,$$

Reciprocal

 * $$\sin{\theta}\,=\frac{1}{\csc{\theta}\,}\,$$


 * $$\cos{\theta}\,=\frac{1}{\sec{\theta}\,}\,$$


 * $$\tan{\theta}\,=\frac{1}{\cot{\theta}\,}\,$$


 * $$\csc{\theta}\,=\frac{1}{\sin{\theta}\,}\,$$


 * $$\sec{\theta}\,=\frac{1}{\cos{\theta}\,}\,$$


 * $$\cot{\theta}\,=\frac{1}{\tan{\theta}\,}\,$$

Periodicity
For the following, let x be any positive whole number.
 * $$\sin{(\theta\pm\,2x\pi)}\,=\sin{\theta}\,$$
 * $$\cos{(\theta\pm\,2x\pi)}\,=\cos{\theta}\,$$
 * $$\tan{(\theta\pm\,x\pi)}\,=\tan{\theta}\,$$
 * $$\csc{(\theta\pm\,2x\pi)}\,=\csc{\theta}\,$$
 * $$\sec{(\theta\pm\,2x\pi)}\,=\sec{\theta}\,$$
 * $$\cot{(\theta\pm\,x\pi)}\,=\cot{\theta}\,$$

Negative Angle

 * $$\sin{-\theta}\,=-\sin{\theta}\,$$
 * $$\cos{-\theta}\,=\cos{\theta}\,$$
 * $$\tan{-\theta}\,=-\tan{\theta}\,$$
 * $$\csc{-\theta}\,=-\csc{\theta}\,$$
 * $$\sec{-\theta}\,=\sec{\theta}\,$$
 * $$\cot{-\theta}\,=-\cot{\theta}\,$$

Cofunction

 * $$\sin{\theta}\,=\cos{(\frac{\pi}{2}\,-\theta)}\,$$


 * $$\cos{\theta}\,=\sin{(\frac{\pi}{2}\,-\theta)}\,$$


 * $$\tan{\theta}\,=\cot{(\frac{\pi}{2}\,-\theta)}\,$$


 * $$\csc{\theta}\,=\sec{(\frac{\pi}{2}\,-\theta)}\,$$


 * $$\sec{\theta}\,=\csc{(\frac{\pi}{2}\,-\theta)}\,$$


 * $$\cot{\theta}\,=\tan{(\frac{\pi}{2}\,-\theta)}\,$$

Pythagorean
The following identities follow a similar form to a2+b2=c2.
 * $$\sin^2{\theta}\,+\cos^2{\theta}\,=1$$
 * $$\tan^2{\theta}\,+1=\sec^2{\theta}\,$$
 * $$\cot^2{\theta}\,+1=\csc^2{\theta}\,$$

Difference and Sum

 * $$\sin{(\alpha+\beta)}\,=\sin{\alpha}\,\cos{\beta}\,+\cos{\alpha}\,\sin{\beta}\,$$
 * $$\sin{(\alpha-\beta)}\,=\sin{\alpha}\,\cos{\beta}\,-\cos{\alpha}\,\sin{\beta}\,$$


 * $$\cos{(\alpha+\beta)}\,=\cos{\alpha}\,\cos{\beta}\,-\sin{\alpha}\,\sin{\beta}\,$$
 * $$\cos{(\alpha-\beta)}\,=\cos{\alpha}\,\cos{\beta}\,+\sin{\alpha}\,\sin{\beta}\,$$


 * $$\tan{(\alpha+\beta)}\,=\frac{\tan{\alpha}\,+\tan{\beta}\,}{1+\tan{\alpha}\,\tan{\beta}\,}\,$$


 * $$\tan{(\alpha-\beta)}\,=\frac{\tan{\alpha}\,-\tan{\beta}\,}{1+\tan{\alpha}\,\tan{\beta}\,}\,$$

Double Angle

 * $$\sin{2\theta}\,=2\sin{\theta}\,\cos{\theta}\,$$


 * $$\cos{2\theta}\,=\cos^2{\theta}\,-\sin^2{\theta}\,$$
 * $$\cos{2\theta}\,=2\cos^2{\theta}\,-1$$
 * $$\cos{2\theta}\,=1-2\sin^2{\theta}\,$$


 * $$\tan{2\theta}\,=\frac{2\tan{\theta}\,}{1-\tan^2{\theta}\,}\,$$

Half-Angle
The plus-minus sign applies for when there is no specified Quadrant. If plus, the angle must land in Quadrants I or IV, where x is positive. If minus, the angle must land in Quadrants II or III, where x is negative.
 * $$\sin{\frac{\theta}{2}\,}\,=\pm\,\sqrt{\frac{1-\cos{\theta}\,}{2}\,}\,$$


 * $$\cos{\frac{\theta}{2}\,}\,=\pm\,\sqrt{\frac{1+\cos{\theta}\,}{2}\,}\,$$


 * $$\tan{\frac{\theta}{2}\,}\,=\pm\,\sqrt{\frac{1-\cos{\theta}\,}{1+\cos{\theta}\,}\,}\,$$