Show that sinh (sinh^-1 x) = x

Answer 1

that is the definition of an inverse function. For any f(x),

f(f^-1(x)) = x
f^-1(f(x)) = x

if f is single-valued, which sinh(x) is.

If you take sinh^-1 of both sides, you have

sinh^-1(sinh(sinh^-1(x))) = sinh^-1(x)
but that only helps if you already know that sinh^-1(sinh(u)) = u.