15.4 Identities

In general, all the trigonometric identities also hold for hyperbolic functions, albeit with minor modifications.

The basic rule of thumb is that wherever you see ±sin2⁡(x)\pm\sin^{2}(x) in "normal" trigonometry, replace it with ∓sinh2⁡(x)\mp\sinh^{2}(x) (so sin2⁡(x)\sin^{2}(x) would be replaced with −sinh2⁡(x)-\sinh^{2}(x), and −sin2⁡(x)-\sin^{2}(x) with sinh2⁡(x)\sinh^{2}(x)). This comes from the relationship sin⁡(i⁢x)=i⁢sinh⁡(x)\sin(ix)=i\sinh(x) which means that sin2⁡(i⁢x)=i2⁢sinh2⁡(x)=−sinh2⁡(x)\sin^{2}(ix)=i^{2}\sinh^{2}(x)=-\sinh^{2}(x). Why this is true was explored in the previous section.

To prove identities, the usual thing to try is to write everything in terms of the exponential function and go from there.

Example 15.4.1

Show that

cosh⁡(2⁢x)=cosh2⁡(x)+sinh2⁡(x)\cosh(2x)=\cosh^{2}(x)+\sinh^{2}(x)

Solution: Usually when proving identities it’s easiest to start with the more "complicated" 66 6 This is a purely qualitative distinction, but it’s usually the side that makes you think either ”yuck” or ”what a fun challenge” depending on your view of mathematics. side.

cosh2⁡(x)+sinh2⁡(x)\displaystyle\cosh^{2}(x)+\sinh^{2}(x) =(e2⁢x+12⁢ex)2+(e2⁢x−12⁢ex)2\displaystyle=\left(\frac{e^{2x}+1}{2e^{x}}\right)^{2}+\left(\frac{e^{2x}-1}{2% e^{x}}\right)^{2} (15.34)
=2⁢e4⁢x+24⁢e2⁢x\displaystyle=\frac{2e^{4x}+2}{4e^{2x}} (15.35)
=e4⁢x+12⁢e2⁢x\displaystyle=\frac{e^{4x}+1}{2e^{2x}} (15.36)
=cosh⁡(2⁢x)\displaystyle=\cosh(2x) (15.37)

Note that for the binomials in Equation 15.34 we know that the middle terms will cancel because the brackets are the same, except for a −1-1 in one and a 11 in the other.