QUESTION: what is cosh x in terms of i as we know that sinh x=1/i(sin ix).
By definition,
cosh(x) = (1/2)(e^x + e^(-x))
and e^(ix) = cos(x) + i.sin(x)
and e^(-ix) = cos(x) - i.sin(x)
so adding the above two equations gives us,
e^(ix) + e^(-ix) = 2.cos(x)
(1/2)(e^(ix) + e^(-ix)) = cos(x)
i.e. cosh(ix) = cos(x)
or, cosh(x) = cos(ix)