Double and Half Angle Formulas for Hyperbolic Functions
📂Functions Double and Half Angle Formulas for Hyperbolic Functions sinh ( 2 x ) = 2 sinh x cosh x cosh ( 2 x ) = cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 2 sinh 2 x + 1 tanh ( 2 x ) = 2 tanh x 1 + tanh 2 x
\begin{align}
\sinh (2x) =&\ 2\sinh x \cosh x \label{1}
\\ \cosh (2x) =&\ \cosh^{2} x + \sinh^{2} x = 2\cosh ^{2 } x -1 = 2\sinh ^{2} x +1
\\ \tanh (2x) =&\ \frac{2\tanh x}{1+\tanh^{2}x}
\end{align}
sinh ( 2 x ) = cosh ( 2 x ) = tanh ( 2 x ) = 2 sinh x cosh x cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 2 sinh 2 x + 1 1 + tanh 2 x 2 tanh x
sinh 2 x 2 = cosh x − 1 2 cosh 2 x 2 = cosh x + 1 2 tanh 2 x 2 = cosh x − 1 cosh x + 1
\begin{align}
\sinh^{2} \frac{x}{2} =&\ \frac{\cosh x -1 }{2}
\\ \cosh^{2} \frac{x}{2} =&\ \frac{\cosh x +1 }{2}
\\ \tanh ^{2} \frac{x}{2} =&\ \frac{\cosh x -1}{\cosh x +1}
\end{align}
sinh 2 2 x = cosh 2 2 x = tanh 2 2 x = 2 cosh x − 1 2 cosh x + 1 cosh x + 1 cosh x − 1
sinh x 2 = sinh x 2 ( cosh x + 1 ) cosh x 2 = sinh x s g n ( x ) 2 ( cosh x − 1 )
\begin{align}
\sinh \frac{x}{2}=&\ \frac{\sinh x}{\sqrt{2(\cosh x +1)}}
\\ \cosh \frac{x}{2}=&\ \frac{\sinh x}{\mathrm{sgn}(x)\sqrt{2(\cosh x -1)}}
\end{align}
sinh 2 x = cosh 2 x = 2 ( cosh x + 1 ) sinh x sgn ( x ) 2 ( cosh x − 1 ) sinh x
s g n ( x ) \mathrm{sgn}(x) sgn ( x ) sign function .
Explanation As we proved the double angle and half angle formulas of trigonometric functions , we use the addition formula of hyperbolic functions for the proof. The process is not difficult. The proof of ( 4 ) − ( 6 ) (4)-(6) ( 4 ) − ( 6 )
Proof Proof of ( 1 ) (1) ( 1 ) sinh ( 2 x ) = sinh ( x + x ) = sinh x cosh x + sinh x cosh x = 2 sinh x cosh x
\begin{align*}
\sinh (2x)=&\ \sinh (x+x)
\\ =&\ \sinh x \cosh x + \sinh x \cosh x
\\ =&\ 2\sinh x \cosh x
\end{align*}
sinh ( 2 x ) = = = sinh ( x + x ) sinh x cosh x + sinh x cosh x 2 sinh x cosh x
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Proof of ( 2 ) (2) ( 2 ) cosh ( 2 x ) = cosh ( x + x ) = cosh x cosh x + sinh x sinh x = cosh 2 x + sinh 2 x
\begin{align*}
\cosh (2x)=&\ \cosh (x+x)
\\ =&\ \cosh x \cosh x + \sinh x \sinh x
\\ =&\ \cosh ^{2} x + \sinh ^{2} x
\end{align*}
cosh ( 2 x ) = = = cosh ( x + x ) cosh x cosh x + sinh x sinh x cosh 2 x + sinh 2 x
Also, since cosh 2 x − sinh 2 x = 1 \cosh^{2} x-\sinh^{2} x=1 cosh 2 x − sinh 2 x = 1
cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 2 sinh x + 1
\cosh^{2} x +\sinh^{2} x = 2\cosh^{2}x -1 = 2\sinh x+1
cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 2 sinh x + 1
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Proof of ( 3 ) (3) ( 3 ) tanh ( 2 x ) = tanh ( x + x ) = sinh ( x + x ) cosh ( x + x ) = 2 sinh x cosh x cosh 2 x + sinh 2 x = 2 sinh x cosh x 1 + sinh 2 x cosh 2 x = 2 tanh x 1 + tanh 2 x
\begin{align*}
\tanh (2x) =\tanh (x+x) =&\ \frac{\sinh(x+x)}{\cosh (x+x)}
\\ =&\ \frac{2\sinh x \cosh x }{\cosh^{2} x + \sinh^{2} x}
\\ =&\ \frac{2\frac{\sinh x}{\cosh x}}{1+\frac{\sinh^{2} x}{\cosh^{2} x}}
\\ =&\ \frac{2\tanh x}{1+\tanh^{2}x}
\end{align*}
tanh ( 2 x ) = tanh ( x + x ) = = = = cosh ( x + x ) sinh ( x + x ) cosh 2 x + sinh 2 x 2 sinh x cosh x 1 + c o s h 2 x s i n h 2 x 2 c o s h x s i n h x 1 + tanh 2 x 2 tanh x
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Proof of ( 7 ) (7) ( 7 ) sinh x 2 = e x − e − x 2 = ( e x 2 − e − x 2 ) ( e x 2 + e − x 2 ) 2 ( e x 2 + e − x 2 ) = e x − e − x 2 1 e x 2 + e − x 2 = sinh x 1 ( e x 2 + e − x 2 ) 2 = sinh x e x + e − x + 2 = sinh x 2 e x + e − x 2 + 2 = sinh x 2 ( cosh x + 1 )
\begin{align*}
\sinh \frac{x}{2} =&\ \frac{e^{x}-e^{-x}}{2}
\\ =&\ \frac{(e^{\frac{x}{2}}-e^{-\frac{x}{2}})(e^{\frac{x}{2}}+e^{-\frac{x}{2}} )}{2(e^{\frac{x}{2}}+e^{-\frac{x}{2}} )}
\\ =&\ \frac{e^{x}-e^{-x}}{2}\frac{1}{e^{\frac{x}{2}}+e^{-\frac{x}{2}}}
\\ =&\ \sinh x \frac{1}{\sqrt{(e^{\frac{x}{2}}+e^{-\frac{x}{2}})^{2}}}
\\ =&\ \frac{\sinh x}{\sqrt{e^{x}+e^{-x}+2}}
\\ =&\ \frac{ \sinh x}{\sqrt{2 \frac{e^{x}+e^{-x}}{2}+2}}
\\ =&\ \frac{\sinh x}{\sqrt{2(\cosh x +1)}}
\end{align*}
sinh 2 x = = = = = = = 2 e x − e − x 2 ( e 2 x + e − 2 x ) ( e 2 x − e − 2 x ) ( e 2 x + e − 2 x ) 2 e x − e − x e 2 x + e − 2 x 1 sinh x ( e 2 x + e − 2 x ) 2 1 e x + e − x + 2 sinh x 2 2 e x + e − x + 2 sinh x 2 ( cosh x + 1 ) sinh x
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Proof of ( 8 ) (8) ( 8 ) cosh x 2 = e x + e − x 2 = ( e x 2 + e − x 2 ) ( e x 2 − e − x 2 ) 2 ( e x 2 − e − x 2 ) = e x − e − x 2 1 e x 2 − e − x 2 = sinh x 1 s g n ( x ) ( e x 2 − e − x 2 ) 2 = sinh x s g n ( x ) e x + e − x − 2 = sinh x s g n ( x ) 2 e x + e − x 2 − 2 = sinh x s g n ( x ) 2 ( cosh x − 1 )
\begin{align*}
\cosh \frac{x}{2} =&\ \frac{e^{x}+e^{-x}}{2}
\\ =&\ \frac{(e^{\frac{x}{2}}+e^{-\frac{x}{2}})(e^{\frac{x}{2}}-e^{-\frac{x}{2}} )}{2(e^{\frac{x}{2}}-e^{-\frac{x}{2}} )}
\\ =&\ \frac{e^{x}-e^{-x}}{2}\frac{1}{e^{\frac{x}{2}}-e^{-\frac{x}{2}}}
\\ =&\ \sinh x \frac{1}{\mathrm{sgn}(x)\sqrt{(e^{\frac{x}{2}}-e^{-\frac{x}{2}})^{2}}}
\\ =&\ \frac{\sinh x}{\mathrm{sgn}(x)\sqrt{e^{x}+e^{-x}-2}}
\\ =&\ \frac{ \sinh x}{\mathrm{sgn}(x)\sqrt{2 \frac{e^{x}+e^{-x}}{2}-2}}
\\ =&\ \frac{\sinh x}{\mathrm{sgn}(x)\sqrt{2(\cosh x -1)}}
\end{align*}
cosh 2 x = = = = = = = 2 e x + e − x 2 ( e 2 x − e − 2 x ) ( e 2 x + e − 2 x ) ( e 2 x − e − 2 x ) 2 e x − e − x e 2 x − e − 2 x 1 sinh x sgn ( x ) ( e 2 x − e − 2 x ) 2 1 sgn ( x ) e x + e − x − 2 sinh x sgn ( x ) 2 2 e x + e − x − 2 sinh x sgn ( x ) 2 ( cosh x − 1 ) sinh x
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