Home » Blog » Prove a formula for sinh(x+y)

Prove a formula for sinh(x+y)

Prove that

    \[ \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y. \]


Proof. We use the definition of the hyperbolic sine in terms of the exponential,

    \begin{align*}  \sinh(x+y) &= \frac{e^{x+y} - e^{-x-y}}{2} \\  &= \frac{2e^{x+y} - 2e^{-x-y}}{4} \\  &= \frac{2e^{x+y} + (e^{x-y} - e^{y-x}) - (e^{x-y} - e^{y-x}) - 2e^{-x}e^{-y}}{4} \\  &= \left( \frac{e^{x+y} + e^{x-y} - e^{-x-y} - e^{y-x}}{4} \right) + \left( \frac{e^{x+y} - e^{x-y} +e^{y-x} -e^{-x-y}}{4} \right) \\  &= \left( \frac{e^x-e^{-x}}{2} \right) \left( \frac{e^y + e^{-y}}{2} \right) + \left( \frac{e^x + e^{-x}}{2} \right) \left( \frac{e^y - e^{-y}}{2} \right) \\  &= \sinh x \cosh y + \cosh x \sinh y. \qquad \blacksquare \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):