Hyperbolic cosine and arctan double angle reductions.  Probably for cosh parameterization of an ellipse.

$Assumptions = m > 0 && phi > 0 && b > 0 && b < 1 && a > 0 && a > b;
Cosh[m+I phi] Cosh[m -I phi]// TrigToExp // FullSimplify

$Assumptions = b > 0 && b < 1 && a > 0 && a > b;
(1 - (b/a)^2)Cosh[2 ArcTanh[b/a]]// FullSimplify

Cosh[2 ArcTanh[b/a]]

1 + Cos[2m] // FullSimplify

1 - 2 Cos[m]Cos[m] // FullSimplify