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\begin{align}
sinh(x)=&\frac{e^x-e^{-x}}{2} \\[1em]
cosh(x)=&\frac{e^x+e^{-x}}{2} \\[1em]
tanh(x)=&\frac{e^x-e^{-x}}{e^x+e^{-x}} \\[1em]
sech(x)=&\frac{2}{e^x+e^{-x}} \\[1em]
csch(x)=&\frac{2}{e^x-e^{-x}} \\[1em]
coth(x)=&\frac{e^x+e^{-x}}{e^x-e^{-x}} \\[0.5em]
\end{align}
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\begin{align}
asinh(x)=&\frac{e^x+e^{-x}}{e^x-e^{-x}} \\[0.5em]
acosh(x)=&\ln{\left( x+\sqrt{x^2-1}\right) } \\[0.5em]
atanh(x)=&\frac{1}{2}\ln{\left( \frac{1+x}{1-x}\right) } \\[0.5em]
asech(x)=&\ln{\left( \frac{1}{x}+\sqrt{\left( \frac{1}{x}\right) ^2-1}\right) } \\[0.5em]
acsch(x)=&\ln{\left( \frac{1}{x}+\sqrt{\frac{1}{x^2+1}}\right) } \\[0.5em]
acoth(x)=&\frac{1}{2}\ln{\left( \frac{x+1}{x-1}\right) } \\[0.5em]
\end{align}
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