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let f(x) = e^(x) / (e^x + e^-x).
let g(x) = 1 / (1 + e^(-2x)).
you need to prove that f(x) = g(x).

e^x / (e^x + e^-x)

e^x / (e^x + 1 / e^x)

e^x / ((e^(2x) + 1) / e^x)

e^(2x) / (e^(2x) + 1). so f(x) = e^(2x) / (e^(2x) + 1)

now take g(x):

1 / (1 + e^(-2x))

1 / (1 + 1/e^(2x))

1 / [ (e^(2x) + 1) / e^(2x) ]

e^(2x) / (e^(2x) + 1). so g(x) = e^(2x) / (e^(2x) + 1).

f(x) = g(x); proved