el7engaginasoulx
Answered

Prove that sinh(x+y)=sinhxcoshy+sinhycoshx. I've tried it from both side. One way I just get ((e^x)(e^y)-(e^-x)(e^-y))/2 and the other way I just get everything to cancel out and I get 1/2..... Help XD

Verified Answer :

caylus

Hello,

\(sh(x)= \dfrac{e^{x}-e^{-x}}{2}\\\\ ch(x)= \dfrac{e^{x}+e^{-x}}{2}\\\\ sh(x)ch(y)+ch(x)sh(y)= \dfrac{e^{x}-e^{-x}}{2}*\frac{e^{y}+e^{-y}}{2}+\frac{e^{x}+e^{-x}}{2}*\frac{e^{y}-e^{-y}}{2}\\\\\)
\(= \dfrac{e^{x+y}-e^{-x+y}+e^{x-y}-e^{-(x+y)}+e^{x+y}+e^{-x+y}-e^{-(x+y)}}{4}\\\\\)
\(=\dfrac{e^{x+y}-e^{-(x+y)}}{2}=sh(x+y)\)

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