Question

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

Answer 1

Hello,


`sh(x)= (e^(x)-e^(-x))/(2)\\\\ ch(x)= (e^(x)+e^(-x))/(2)\\\\ sh(x)ch(y)+ch(x)sh(y)= (e^(x)-e^(-x))/(2)*(e^(y)+e^(-y))/(2)+(e^(x)+e^(-x))/(2)*(e^(y)-e^(-y))/(2)\\\\`

`= (e^(x+y)-e^(-x+y)+e^(x-y)-e^(-(x+y))+e^(x+y)+e^(-x+y)-e^(-(x+y)))/(4)\\\\`

`=(e^(x+y)-e^(-(x+y)))/(2)=sh(x+y)`