cosh x = e x + e − x 2 sinh x = e x − e − x 2 tanh x = sinh x cosh x = e x − e − x e x + e − x sin 2 x + cos 2 x = 1 = tanh 2 x + sec h 2 x cosh 2 x = ( e x + e − x 2 ) ⋅ ( e x + e − x 2 ) = e 2 x + e − 2 x + 2 4 sinh 2 x = e 2 x + e − 2 x − 2 4 → cosh 2 x − sinh 2 x = 1 {\displaystyle {\begin{aligned}&\cosh x={\frac {e^{x}+e^{-x}}{2}}\\&\sinh x={\frac {e^{x}-e^{-x}}{2}}\\&\tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}\\&\sin ^{2}x+\cos ^{2}x=1=\tanh ^{2}x+\sec h^{2}x\\&\cosh ^{2}x=\left({\frac {e^{x}+e^{-x}}{2}}\right)\cdot \left({\frac {e^{x}+e^{-x}}{2}}\right)={\frac {e^{2x}+e^{-2x}+2}{4}}\\&\sinh ^{2}x={\frac {e^{2x}+e^{-2x}-2}{4}}\to \cosh ^{2}x-\sinh ^{2}x=1\\\end{aligned}}}
