# Đẳng thức lượng giác

## Hàm số

${\displaystyle e^{x}=\sinh x+\cosh x}$
${\displaystyle \cosh ^{2}x-\sinh ^{2}x=1}$
${\displaystyle \mathrm {sech} ^{2}x=1-\tanh ^{2}x}$
${\displaystyle \mathrm {csch} ^{2}x=\mathrm {coth} ^{2}x-1}$
${\displaystyle \sinh x=-i\sin ix={\frac {e^{x}-e^{-x}}{2}}}$
${\displaystyle \cosh x=\cos ix={\frac {e^{x}+e^{-x}}{2}}}$
${\displaystyle \tanh x=-i\tan ix={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}$
${\displaystyle \mathrm {csch} x=i\csc ix={\frac {2}{e^{x}-e^{-x}}}}$
${\displaystyle \mathrm {sech} x=\sec ix={\frac {2}{e^{x}+e^{-x}}}}$
${\displaystyle \mathrm {coth} x=i\cot ix={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}}$

## Đẳng thức hàm số nghịch

${\displaystyle \mathrm {arsinh} x=\int _{0}^{x}{\frac {1}{\sqrt {t^{2}+1}}}\mathrm {d} t=\log \left(x+{\sqrt {x^{2}+1}}\right)}$
${\displaystyle \mathrm {arcosh} x=\int _{1}^{x}{\frac {1}{\sqrt {t^{2}-1}}}\mathrm {d} t=\log \left(x+{\sqrt {x^{2}-1}}\right)}$
${\displaystyle \mathrm {artanh} x=\int _{0}^{x}{\frac {1}{1-t^{2}}}\mathrm {d} t={\frac {1}{2}}\log \left({\frac {1+x}{1-x}}\right)}$
${\displaystyle \mathrm {arccsh} x=\log \left({\frac {1+{\sqrt {1+x^{2}}}}{x}}\right)}$
${\displaystyle \mathrm {arsech} x=\log \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)}$
${\displaystyle \mathrm {arcoth} x={\frac {1}{2}}\log \left({\frac {x+1}{x-1}}\right)}$