Trigonometric Identities

Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form.

Basic Identities:

$$\sin(x)=\frac{1}{\csc(x)}$$

$$\cos(x)=\frac{1}{\sec(x)}$$

$$\tan(x)=\frac{1}{\cot(x)}$$

$$\sec(x)=\frac{1}{\cos(x)}$$

$$\csc(x)=\frac{1}{\sin(x)}$$

$$\cot(x)=\frac{1}{\tan(x)}$$

$$\tan(x)=\frac{\sin(x)}{\cos(x)}$$

$$\sin(-x)=-\sin(x)$$

$$\cos(-x)=\cos(x)$$

$$\tan(-x)=-\tan(x)$$

Pythagorean Identities

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

$$1+\tan^2(x)=\sec^2(x)$$

$$1+\cot^2(x)=\csc^2(x)$$

Sum and Difference Formulas

$$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$$

$$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$$

$$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$$

$$\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$$

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

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

$$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$$

$$\sin(x)-\sin(y)=2\cos(\frac{x+y}{2})\sin(\frac{x-y}{2})$$

$$\cos(x)+\cos(y)=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})$$

$$\cos(x)-\cos(y)=-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})$$

Double Angle Formulas

$$\sin(2x)=2\sin(x)\cos(x)$$

$$\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1$$

Half Angle Formulas

$$\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}$$

$$\cos(\frac{x}{2})=\pm\sqrt{\frac{1+\cos(x)}{2}}$$

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

Trigonometric Products

$$\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}$$

$$\cos(x)\cos(y)=\frac{\cos(x+y)+\cos(x-y)}{2}$$

$$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}$$