ch\left({α±ß}\right)=ch\ α·ch\ ß±sh\ α·sh\ ß
sh\left({α±ß}\right)=sh\ α·ch\ ß±ch\ α·sh\ ß
th\left({α±ß}\right)=\frac{th\ α±th\ ß}{1±th\ α·th\ ß}
cth\left({α±ß}\right)=\frac{1±cth\ α·cth\ ß}{cth\ α±cth\ ß}
\begin{array}{rccccl} ch^2\ α-sh^2\ α = 1 & = & \left({ch\ α + sh\ α}\right) & · & \left({ch\ α - sh\ α}\right) & =\\ & {\ } &|| & {\ } &|| & {\ } \\ & = & e^α \quad & · & e^{-α} & = \displaystyle{\frac{e^α}{e^α}}=1 \end{array}
USEFUL AjAx LaTeX REFERENCES (to produce the above) ::
https://docs.mathjax.org/en/v2.7-latest/tex.html
https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference
ch^2\ α+sh^2\ α=ch\left({2α}\right)=2ch^2\ α-1=2sh^2\ α+1
sh\left({2α}\right)=2·sh\ α·ch\ α
th\left({2α}\right)=\frac{2·th\ α}{1+th^2\ α}
cth\left({2α}\right)=\frac{1+cth^2\ α}{2·cth\ α}
1-th^2\ α=\frac1{ch^2\ α}
1-cth^2\ α=\frac{-1}{sh^2\ α}
th\frac α2=\frac{ch\ α-1}{sh\ α}=\frac{sh\ α}{ch\ α+1}
cth\frac α2=\frac{sh\ α}{ch\ α-1}=\frac{ch\ α+1}{sh\ α}
ch\ α±ch\ ß=2·{}^{ch}_{sh}\frac{α+ß}2·{}^{ch}_{sh}\frac{α-ß}2
sh\ α±sh\ ß=2·sh\frac{α±ß}2·ch\frac{α∓ß}2
\displaystyle{th\ α±th\ ß=\frac{sh\left({α±ß}\right)}{ch\ α·ch\ ß}}
\displaystyle{cth\ α±cth\ ß=\frac{sh\left({ß±α}\right)}{sh\ ß·sh\ α}}
arch\ A=ln\left({A±\sqrt{A^2-1}}\right)
arsh\ A=ln\left({A±\sqrt{A^2+1}}\right)
arth\ A=\frac12·ln\frac{1+A}{1-A}
arcth\ A=\frac12·ln\frac{A+1}{A-1}
arth\ A+arcth\ A=ln\ i+ln\frac{A+1}{A-1}=ln\ i+ln\frac{1+A}{1-A}
±\left({arth\ A-arcth\ A}\right)=ln\ i
[Eop]
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