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Prove that sinh(2x) = 2 sinh x cosh x

Prove that

    \[ \sinh (2x) = 2 \sinh x \cosh x. \]


Proof. We know from this exercise (Section 6.19, Exercise #5) that

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

Therefore,

    \begin{align*}  \sinh (2x) &= \sinh (x+x) \\  &= \sinh x \cosh x + \cosh x \sinh x \\  & = 2 \sinh x \cosh x. \qquad \blacksquare \end{align*}

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