Trigonometry/Trigonometric Identities Reference

Pythagoras

1. ${\sin }^2(x)+{\cos }^2(x)=1$
2. $1+{\tan }^2(x)={\sec }^2(x)$
3. $1+{\cot }^2(x)={\csc }^2(x)$

These are all direct consequences of Pythagoras's theorem.

Sum/Difference of angles

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

Product to Sum

1. $2\sin (x)\sin (y)=\cos (x-y)-\cos (x+y)$
2. $2\cos (x)\cos (y)=\cos (x-y)+\cos (x+y)$
3. $2\sin (x)\cos (y)=\sin (x-y)+\sin (x+y)$

Sum and difference to product

1. $A\sin (x)+B\cos (x)=C\sin (x+y)$, where $C=\sqrt{A^2+B^2}$ and $y=\pm \arctan (B/A)$
2. $\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$
3. $\sin \alpha -\sin \beta =2\cos \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$
4. $\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$
5. $\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$

Double angle

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

These are all direct consequences of the sum/difference formulae

Half angle

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

In cases with $\pm$, the sign of the result must be determined from the value of $\frac{x}{2}$. These derive from the $\cos (2x)$ formulae.

Power Reduction

1. ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$
2. ${\cos }^2\theta =\frac{1+\cos 2\theta }{2}$
3. ${\tan }^2\theta =\frac{1-\cos 2\theta }{1+\cos 2\theta }$

Even/Odd

1. $\sin (-\theta )=-\sin (\theta )$
2. $\cos (-\theta )=\cos (\theta )$
3. $\tan (-\theta )=-\tan (\theta )$
4. $\csc (-\theta )=-\csc (\theta )$
5. $\sec (-\theta )=\sec (\theta )$
6. $\cot (-\theta )=-\cot (\theta )$

Calculus

1. $\frac{d}{dx}[\sin x]=\cos x$
2. $\frac{d}{dx}[\cos x]=-\sin x$
3. $\frac{d}{dx}[\tan x]={\sec }^{2}x$
4. $\frac{d}{dx}[\sec x]=\sec x\tan x$
5. $\frac{d}{dx}[\csc x]=-\csc x\cot x$
6. $\frac{d}{dx}[\cot x]=-{\csc }^{2}x$