[tex]x'=x\cosh{\omega}-ct\sinh{\omega}[/tex]

[tex]t'=-x\sinh{\omega}+ct\cosh{\omega}[/tex]

is used for the lorentz transformations, where the velocity is [tex]v=\frac{c\sinh{\omega}}{\cosh{\omega}}[/tex].

However, I'm wondering, couldn't you also do

[tex]x'=x\sec{\omega}-ct\tan{\omega}[/tex]

[tex]t'=-x\tan{\omega}+ct\sec{\omega}[/tex]

(or even the similar thing with [tex]\csc{\omega}[/tex] and [tex]\cot{\omega}[/tex])

With [tex]v=\frac{c\tan{\omega}}{\sec{\omega}}[/tex]

Since this also reproduces the lorentz transformations

Is there any advantage to using the hyperbolic functions instead?