List of trigonometric identities

Fundamental Identities

Quotient and reciprocal identities

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

Pythagorean identities

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

Angle sum and difference identities

• $\sin(x\pm y)=\sin(x)\cos(y)\pm\sin(y)\cos(x)$
• $\cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y)$
• $\tan(x\pm y)=\frac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)}$

Double-angle identities

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

Half-angle identities

• $\sin\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos(x)}{2}}$
• $\cos\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1+\cos(x)}{2}}$
• $\tan\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\frac{\sin(x)}{1+\cos(x)}=\frac{1-\cos(x)}{\sin(x)}$