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Trigonometric Rules

$\displaystyle\tan x={{\sin x}\over {\cos x}}$ , $\displaystyle\cot x={{\cos x}\over {\sin x}}$

$\displaystyle\sec x={1\over {\cos x}}$ , $\displaystyle\csc x={1\over {\sin x}}$

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

$\sin(x\pm y)=\sin x\cdot\cos y\pm\cos x\cdot\sin y$

$\cos(x\pm y)=\cos x\cdot\cos y\mp\sin x\cdot\sin y$

$\displaystyle\displaystyle\tan(x\pm y)={{\tan x\pm\tan y}\over {1\mp\tan x\cdot\tan y}}$

$\displaystyle\cot(x\pm y)={{\cot y\cdot\cot x\mp 1}\over {\cot y\pm\cot x}}$

$\displaystyle\sin x\pm\sin y=2\sin{{x\pm y}\over 2}\cdot\cos{{x\mp y}\over 2}$

$\displaystyle\cos x+\cos y=2\cos{{x+y}\over 2}\cdot\cos{{x-y}\over 2}$

$\displaystyle\cos x-\cos y=-2\sin{{x+y}\over 2}\cdot\sin{{x-y}\over 2}$

$\displaystyle\tan x\pm\tan y={{\sin(x\pm y)}\over {\cos x\cdot\cos y}}$

$\displaystyle\cot x\pm\cot y={{\sin(x\pm y)}\over {\sin x\cdot\sin y}}$

$\sin2x=2\sin x\cdot\cos x$

$\cos2x=\cos^2-\sin^2x$

$\displaystyle\tan2x={{2\tan x}\over {1-\tan^2x}}$

$\displaystyle\cot2x={{\cot^2x-1}\over {2\cot x}}$

$\displaystyle\sin^2{x\over 2}={{1-\cos x}\over 2}$

$\displaystyle\cos^2{x\over 2}={{1+\cos x}\over 2}$

$\displaystyle\tan{x\over 2}={{1-\cos x}\over {\sin x}}={{\sin x}\over {1+\cos x}}$

Right triangle

c²=a²+b²

$a=c\sin\alpha$

$b=c\cos\alpha$

Law of sines $\displaystyle{{a\over {\sin\alpha}}}={b\over {\sin\beta}}={c\over {\sin\gamma}}=2r$

Law of cosines $c^2=a^2+b^2-2ab\cos\gamma$