# Trigonometry/For Enthusiasts/Less-Used Trig Identities

## Triangle IdentitiesEdit

In addition to the Law of Sines, the Law of Cosines, and the Law of Tangents, there are numerous other identities that apply to the three angles A, B, and C of any triangle (where A+B+C=180° and each of A, B, and C is greater than zero). Some of the most notable ones follow:

1. $\displaystyle \cos^2A+\cos^2B+\cos^2C+2\cos A \cos B \cos C=1$
2. $\sin A + \sin B + \sin C = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$
3. $\displaystyle \tan A + \tan B + \tan C = \tan A \tan B \tan C$
4. $\tan \frac{A}{2}\tan\frac{B}{2} + \tan\frac{B}{2}\tan\frac{C}{2} + \tan\frac{C}{2}\tan\frac{A}{2} = 1$
5. $\displaystyle \cot A \cot B + \cot B \cot C + \cot C \cot A = 1$
6. $\cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2} = \cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}$
7. $\sin A \sin B \sin C = \frac{1}{(\cot A + \cot B)(\cot B + \cot C)(\cot C + \cot A)}$
8. $\frac{\sin A + \sin B - \sin C}{\sin A + \sin B + \sin C} = \tan \frac{A}{2}\tan \frac{B}{2}$

## PythagorasEdit

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

These are all direct consequences of Pythagoras's theorem.

## Sum/Difference of anglesEdit

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 SumEdit

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

## Sum and difference to productEdit

1. $\displaystyle A \sin(x)+B\cos(x)= C \sin(x+y)$, 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. $\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$

## Multiple angleEdit

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. $\tan(2x)=\frac{2\tan(x)}{1- \tan^2(x)}$
4. $\cot(2x)=\frac{1}{2}[\cot(x)-\tan(x)]$
5. $\csc(2x)=\frac{1}{2}[\cot(x)+\tan(x)]$
6. $\cos(3x)=4\cos^{3}(x)-3\cos(x)$
7. $\sin(3x)=-4\sin^{3}(x)+3\sin(x)$
8. $\tan(3x)=\frac{3\tan(x)-\tan^{3}(x)}{1-3\tan^{2}(x)}$
9. $\cos(4x)=8\cos^{4}(x)-8\cos^{2}(x)+1$
10. $\sin(4x)=4\sin(x)\cos^{3}(x)-4\sin^{3}(x)\cos(x)$
11. $\sin^{2}(4x)=16[\sin^{2}(x)-5\sin^{4}(x)+8\sin^{6}(x)-4\sin^{8}(x)]$
12. $\tan(4x)=\frac{4\tan(x)-4\tan^{3}(x)}{1-6\tan^{2}(x)+\tan^{4}(x)}$
13. $\cos(5x)=16\cos^{5}(x)-20\cos^{3}(x)+5\cos(x)$
14. $\sin(5x)=16\sin^{5}(x)-20\sin^{3}(x)+5\sin(x)$
15. $\tan(5x)=\frac{5\tan(x)-10\tan^{3}(x)+\tan^{5}(x)}{1-10\tan^{2}(x)+5\tan^{4}(x)}$
16. $\cos(6x)=32\cos^{6}(x)-48\cos^{4}(x)+18\cos^{2}(x)-1$
17. $\cos(7x)=64\cos^{7}(x)-112\cos^{5}(x)+56\cos^{3}(x)-7\cos(x)$
18. $\sin(7x)=-64\sin^{7}(x)+112\sin^{5}(x)-56\sin^{3}(x)+7\sin(x)$
19. $\cos(8x)=128\cos^{8}(x)-256\cos^{6}(x)+160\cos^{4}(x)-32\cos^{2}(x)+1$
20. $\cos(nx)=2\cos(x)\cos[(n-1)x]-\cos[(n-2)x]$
21. $\sin(nx)=2\cos(x)\sin[(n-1)x]-\sin[(n-2)x]$

These are all direct consequences of the sum/difference formulae

## Half angleEdit

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

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 ReductionEdit

1. $\sin^2\theta=\frac{1-\cos2\theta}{2}$
2. $\cos^2\theta=\frac{1+\cos2\theta}{2}$
3. $\tan^2\theta=\frac{1-\cos2\theta}{1+\cos2\theta}$

## Even/OddEdit

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. $\cot(-\theta)=-\cot(\theta)$

## CalculusEdit

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. $\frac{d}{dx}[\cot x] = -\csc^{2} x$