Trigonometric identities

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1: Geometric proof of the sum and difference of two angles

Trigonometric identities are fundamental identities in mathematics involving trigonometric functions. Here are some common trigonometric identities:

basic identity

1. ${\sin }^{2}x+{\cos }^{2}x=1$
2. $\tan x=\frac{\sin x}{\cos x}$
3. $\cot x=\frac{\cos x}{\sin x}$

sum and difference

1. $\sin (x+y)=\sin x\cos y+\cos x\sin y$
2. $\sin (x-y)=\sin x\cos y-\cos x\sin y$
3. $\cos (x+y)=\cos x\cos y-\sin x\sin y$
4. $\cos (x-y)=\cos x\cos y+\sin x\sin y$
5. $\mathrm{so}(x)+y)=\frac{\tan x+tany}{1-\tan x\tan y}$
6. $\mathrm{so}(x)-y)=\frac{\tan x-tany}{1+\tan x\tan y}$

double angle identity

1. $\sin 2x\_=2\sin x\cos x$
2. $\cos 2x\_={\cos }^{2}x-{\sin }^{2}x$
3. $\cos 2x\_=2{\cos }^{2}x-1$
4. $\cos 2x\_=1-2{\sin }^{2}x$
5. $\tan 2x\_=\frac{2\tan x}{1-{\tan }^{2}x}$

half angle identity

1. ${\sin }^{2}x=\frac{1-\cos 2}{2}$
2. ${\cos }^{2}x=\frac{1+\cos 2}{2}$

product to sum

1. $\sin x\sin y=\frac{1}{2}[\cos (x-y)-\cos (x+y)]$
2. $\cos x\cos y=\frac{1}{2}[\cos (x-y)+\cos (x+y)]$
3. $\sin x\cos y=\frac{1}{2}[\sin (x+y)+\sin (x-y)]$

These identities are very useful in solving trigonometry problems and are the basis for many problems in mathematics, physics, and engineering.