Sum of Hyperbolic Tangent and Cotangent

Theorem

$\tanh x + \coth x = 2 \coth 2 x$

Proof

\(\ds \tanh x + \coth x\)

\(=\)

\(\ds \frac {\sinh x} {\cosh x} + \frac {\cosh x} {\sinh x}\)

Definition of Hyperbolic Tangent and Hyperbolic Cotangent

\(\ds \)

\(=\)

\(\ds \frac {\cosh^2 x + \sinh^2 x} {\sinh x \cosh x}\)

\(\ds \)

\(=\)

\(\ds 2 \frac {\cosh^2 x + \sinh^2 x} {2 \sinh x \cosh x}\)

\(\ds \)

\(=\)

\(\ds 2 \frac {\cosh 2 x} {\sinh 2 x}\)

Double Angle Formula for Hyperbolic Sine and Double Angle Formula for Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds 2 \coth 2 x\)

Definition of Hyperbolic Cotangent

$\blacksquare$