Originally Posted by
ticbol 1.) sinh(x+y) = sinh(x)cosh(y) +cosh(x)sin(y)
I have done this before.
I don't know how link, so here is a "copy-paste" of it:
----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.
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