Hyperbolic Functions and the Hyperbola

If we plot (cos(t), sin(t)) and allow t to vary, we parameterize the unit circle
x² + y² = 1.
If we plot (cosh(t), sinh(t)) and allow t to vary, we see that these points parameterize the parabola x² - y² = 1. This is a consequence of the identity
cosh²(x) - sinh²(x) = 1.

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Drag the slider to change the value of t.
Click the checkbox to show the parabola x² + y² = 1.

Jason McCullough, Created with GeoGebra