Hyperbolic functions — common mistakes

Be very careful with the identities. The minus sign in $\cosh^2(x) - \sinh^2(x) = 1$ is a common source of errors. Always double-check if you are using a hyperbolic or a trigonometric identity. When sketching $y=\cosh(x)$, ensure your minimum is at (0,1) and the curve is U-shaped, but distinct from a parabola (it's steeper).

They are called hyperbolic because they parametrise the unit hyperbola $x^2 - y^2 = 1$ with the point $(\cosh(t), \sinh(t))$, in the same way that trigonometric functions parametrise the unit circle $x^2 + y^2 = 1$ with the point $(\cos(t), \sin(t))$.

Although they look similar near the origin, they are different curves. The catenary, $y = \cosh(x)$, grows exponentially, so it becomes much steeper than the parabola $y = x^2 + 1$ as $x$ moves away from zero. A parabola is a quadratic function, while a catenary is based on exponential functions.

Yes, it is highly recommended. Exam questions can explicitly ask you to prove an identity from the definitions, as shown in the worked example. Understanding the proofs also deepens your grasp of the relationship between hyperbolic and exponential functions.

If the equation contains a mix of different hyperbolic functions (e.g., $\sinh(x)$ and $\cosh(x)$), using the exponential definitions is often the most direct route. If the equation involves powers of a single function (e.g., $\cosh^2(x)$) and another function, try using an identity like $\sinh^2(x) = \cosh^2(x) - 1$ to create a polynomial in one hyperbolic function.