Which hyperbolic trigonometric identity is the analogue to the circular identity cos^2(x) + sin^2(x) = 1?

cosh^2(x) - sinh^2(x) = 1

While circular functions are defined relative to the unit circle (x^2 + y^2 = 1), hyperbolic functions are defined relative to the unit hyperbola (x^2 - y^2 = 1), leading to this sign difference in the identity.