Sum of Squares of Hyperbolic Secant and Tangent

Theorem

$\sech^2 x + \tanh^2 x = 1$

where $\sech$ and $\tanh$ are hyperbolic secant and hyperbolic tangent.

Proof

\(\ds \sech^2 x + \tanh^2 x\)

\(=\)

\(\ds \paren {\frac 2 {e^x + e^{-x} } }^2 + \tanh^2 x\)

Definition 1 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \paren {\frac 2 {e^x + e^{-x} } }^2 + \paren {\frac {e^x - e^{-x} } {e^x + e^{-x} } }^2\)

Definition 1 of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac {4 + e^{2 x} - 2 + e^{-2 x} } {e^{2 x} + 2 + e ^{-2 x} }\)

Exponent Combination Laws

\(\ds \)

\(=\)

\(\ds \frac {e^{2 x} + 2 + e ^{-2 x} } {e^{2 x} + 2 + e ^{-2 x} }\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Also presented as

Sum of Squares of Hyperbolic Secant and Tangent can also be reported as:

$1 - \tanh^2 x = \sech^2 x$

or:

$1 - \sech^2 x = \tanh^2 x$

Also see

• Difference of Squares of Hyperbolic Cotangent and Cosecant

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.12$: Relationships among Hyperbolic Functions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions