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      <text>6.2.2.7 Common hyperbolic identities</text>
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<h4>Key Point</h4>
<p>The fundamental identity relating hyperbolic functions is:\[\cosh^2x - \sinh^2x \equiv 1.\] This is the hyperbolic function equivalence of the trigonometric identity: \(\cos^2x + \sin^2x \equiv 1\).</p></div><br><br>
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<h4>Key Point: More hyperbolic identities</h4>
\[\begin{align*} \cosh(x+y) &amp;\equiv \cosh x \cosh y+\sinh x \sinh y \\ \sinh(x+y) &amp;\equiv \sinh x \cosh y+\cosh x\sinh y \\ \sinh2x &amp;\equiv 2\sinh x \cosh x \\ \cosh2x &amp;\equiv \cosh^2x+\sinh^2x \equiv 2\cosh^2x-1 \equiv 1+2\sinh^2x \end{align*} \]
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