----sinh u = (1/2)[e^u -e^(-u)]

----cosh u = (1/2)[e^u +e^(-u)]

----a^b * a^c = a^(b+c)

sinh(x+y) = sinh x cosh y + cosh x sinh y

We develop the RHS.

RHS =

= {(1/2)[e^x -e^(-x)] *(1/2)[e^y +e^(-y)]} +{(1/2)[e^x +e^(-x)] *(1/2)[e^y -e^(-y)]}

= (1/4)[e^x -e^(-x)][e^y +e^(-y)] +(1/4)[e^x +e^(-x)][e^y -e^(-y)]

Doing FOILs in the expansion,

= (1/4)[e^(x+y) +e^(x-y) -e^(-x+y) -e^(-x-y)] +(1/4)[e^(x+y) -e^(x-y) +e^(-x+y) -e^(-x-y)]

= (1/4)[e^(x+y) +e^(x-y) -e^(-x+y) -e^(-x-y) +e^(x+y) -e^(x-y) +e^(-x+y) -e^(-x-y)]

= (1/4)[2e^(x+y) -2e^(-x-y)]

= (1/2)[e^(x+y) -e^(-x-y)]

= (1/2)[e^(x+y) -e^[-(x+y)]

= sinh (x+y)

= LHS

Therefore, proven.