\(\operatorname{sech}(x)\) is defined as:
\[\operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \]
If we square this:
\[\operatorname{sech}^2(x) = \frac{4}{e^{2x} + 2 + e^{-2x}} \]
\(\tanh(x)\) is defiend as:
\[\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]
If we square this:
\[ \tanh^2(x) = \frac{e^{2x} -2+ e^{-2x}}{e^{2x} +2+ e^{-2x}} \]
If we add \(\tanh^2(x)\) and \(\operatorname{sech}^2(x)\):
\[ \tanh^2(x) + \operatorname{sech}^2(x) = \frac{e^{2x} -2+ e^{-2x}}{e^{2x} +2+ e^{-2x}} + \frac{4}{e^{2x} + 2 + e^{-2x}} = \frac{e^{2x} +2+ e^{-2x}}{e^{2x} +2+ e^{-2x}} = 1 \]
This means \(\operatorname{sech}^2(x) = 1 - \tanh^2(x)\)