\(\bullet \;\displaystyle \sinh x =\frac{e^x − e^{−x}}{2}\)
\(\bullet \;\displaystyle \cosh x =\frac{e^x + e^{−x}}{2}\)
\(\bullet \;\displaystyle \sin x =\frac{e^{ix} - e^{−ix}}{2i}\)
\(\bullet \;\displaystyle \cos x =\frac{e^{ix} + e^{−ix}}{2}\)
\(\bullet \;\displaystyle \tanh x =\frac{e^x - e^{−x}}{e^x + e^{−x}}\)
\(\bullet \;\displaystyle \tanh x =\frac{\sinh x}{\cosh x}\)
\(\bullet \;\displaystyle \coth x =\frac{\cosh x}{\sinh x}\)
\(\bullet \;\displaystyle \mathrm{sech}(x) =\frac{1}{\cosh x}\)
\(\bullet \;\displaystyle \mathrm{csch}(x) =\frac{1}{\sinh x}\)
\(\bullet \;\displaystyle \sinh (−x) = − \sinh x\)
\(\bullet \;\displaystyle \cosh (−x) = \cosh x\)
\(\bullet \;\displaystyle \tanh (−x) = − \tanh x \)
\(\bullet \;\displaystyle \coth (−x) = − \coth x\)
\(\bullet \;\displaystyle \mathrm{sech}(-x) =\mathrm{sech}(x)\)
\(\bullet \;\displaystyle \mathrm{csch}(-x) =-\mathrm{csch}(x)\)
\(\bullet \;\displaystyle \cosh^{2} x− \sinh^{2} x = 1\)
\(\bullet \;\displaystyle \tanh^{2} x+ \mathrm{sech^{2}}(x) = 1 \)
\(\bullet \displaystyle \sinh (x + y) = \sinh x \cosh y + \cosh x \sinh y\)
\(\bullet \displaystyle \cosh (x + y) = \cosh x \cosh y+ \sinh x \sinh y\)
\(\bullet \displaystyle \sinh (x − y) = \sinh x \cosh y − \cosh x \sinh y\)
\(\bullet \displaystyle \cosh (x − y) = \cosh x \cosh y− \sinh x \sinh y\)
\(\bullet \;\displaystyle \tanh (x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)
\(\bullet \;\displaystyle \sinh 2x = 2 \sinh x \cosh x\)
\(\bullet \;\displaystyle \cosh 2x = \cosh^{2} x+ \sinh^{2} x\)
\(\bullet \;\displaystyle \cosh^{2} x =\frac{1 + \cosh 2x}{2}\)
\(\bullet \;\displaystyle \sinh^{2} x =\frac{\cosh 2x − 1}{2}\)
\(\bullet \;\displaystyle \tanh 2x =\frac{2 \tanh x}{1+ \tanh^{2} x}\)
\(\bullet \;\displaystyle \sinh^{2} x =\frac{\cosh 2x − 1}{2}\)
\(\bullet \;\displaystyle \mathrm{arcsinh} x = \ln \left (x + \sqrt{x^2 + 1} \right ) \)
\(\bullet \;\displaystyle \mathrm{arccosh} x= \ln \left ( {x + \sqrt{x^2 - 1}} \right ) \)
\(\bullet \;\displaystyle \mathrm{arctanh} x = \ln \left (\sqrt{\frac{1 + x}{1 - x}} \right ) \)
\(\bullet \;\displaystyle\mathrm{arccoth} x = \ln \left ( \sqrt{\frac{x + 1}{x - 1}} \right )\)
\(\bullet \;\displaystyle \mathrm{arcsech} x = \ln \left ( \frac{1 + \sqrt{1 - x^2}}{x} \right )\)
\(\bullet \;\displaystyle \mathrm{arccsch} x = \ln \left (\frac{{1 + \sqrt{1 + x^2}}}{|x|} \right )\)
\(\bullet \;\displaystyle \cosh x =\frac{e^x + e^{−x}}{2}\)
\(\bullet \;\displaystyle \sin x =\frac{e^{ix} - e^{−ix}}{2i}\)
\(\bullet \;\displaystyle \cos x =\frac{e^{ix} + e^{−ix}}{2}\)
\(\bullet \;\displaystyle \tanh x =\frac{e^x - e^{−x}}{e^x + e^{−x}}\)
\(\bullet \;\displaystyle \tanh x =\frac{\sinh x}{\cosh x}\)
\(\bullet \;\displaystyle \coth x =\frac{\cosh x}{\sinh x}\)
\(\bullet \;\displaystyle \mathrm{sech}(x) =\frac{1}{\cosh x}\)
\(\bullet \;\displaystyle \mathrm{csch}(x) =\frac{1}{\sinh x}\)
\(\bullet \;\displaystyle \sinh (−x) = − \sinh x\)
\(\bullet \;\displaystyle \cosh (−x) = \cosh x\)
\(\bullet \;\displaystyle \tanh (−x) = − \tanh x \)
\(\bullet \;\displaystyle \coth (−x) = − \coth x\)
\(\bullet \;\displaystyle \mathrm{sech}(-x) =\mathrm{sech}(x)\)
\(\bullet \;\displaystyle \mathrm{csch}(-x) =-\mathrm{csch}(x)\)
\(\bullet \;\displaystyle \cosh^{2} x− \sinh^{2} x = 1\)
\(\bullet \;\displaystyle \tanh^{2} x+ \mathrm{sech^{2}}(x) = 1 \)
\(\bullet \displaystyle \sinh (x + y) = \sinh x \cosh y + \cosh x \sinh y\)
\(\bullet \displaystyle \cosh (x + y) = \cosh x \cosh y+ \sinh x \sinh y\)
\(\bullet \displaystyle \sinh (x − y) = \sinh x \cosh y − \cosh x \sinh y\)
\(\bullet \displaystyle \cosh (x − y) = \cosh x \cosh y− \sinh x \sinh y\)
\(\bullet \;\displaystyle \tanh (x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)
\(\bullet \;\displaystyle \sinh 2x = 2 \sinh x \cosh x\)
\(\bullet \;\displaystyle \cosh 2x = \cosh^{2} x+ \sinh^{2} x\)
\(\bullet \;\displaystyle \cosh^{2} x =\frac{1 + \cosh 2x}{2}\)
\(\bullet \;\displaystyle \sinh^{2} x =\frac{\cosh 2x − 1}{2}\)
\(\bullet \;\displaystyle \tanh 2x =\frac{2 \tanh x}{1+ \tanh^{2} x}\)
\(\bullet \;\displaystyle \sinh^{2} x =\frac{\cosh 2x − 1}{2}\)
\(\bullet \;\displaystyle \mathrm{arcsinh} x = \ln \left (x + \sqrt{x^2 + 1} \right ) \)
\(\bullet \;\displaystyle \mathrm{arccosh} x= \ln \left ( {x + \sqrt{x^2 - 1}} \right ) \)
\(\bullet \;\displaystyle \mathrm{arctanh} x = \ln \left (\sqrt{\frac{1 + x}{1 - x}} \right ) \)
\(\bullet \;\displaystyle\mathrm{arccoth} x = \ln \left ( \sqrt{\frac{x + 1}{x - 1}} \right )\)
\(\bullet \;\displaystyle \mathrm{arcsech} x = \ln \left ( \frac{1 + \sqrt{1 - x^2}}{x} \right )\)
\(\bullet \;\displaystyle \mathrm{arccsch} x = \ln \left (\frac{{1 + \sqrt{1 + x^2}}}{|x|} \right )\)
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