Simplify and FullSimplify do not simplify a simple term Why don't Simplify and FullSimplify work on:  $\frac{\sqrt{a + \cos \theta}}{\sqrt{\frac{a + \cos \theta}{1 + a}}}$?
FullSimplify[ Sqrt[a + Cos[\[Theta]]] /Sqrt[(a + Cos[\[Theta]])/(
 1 + a)]]

Even if $\cos \theta \leq 0$ the term should simplify.... no?

You need to give assumptions, try this:
    FullSimplify[ Sqrt[a + Cos[\[Theta]]] /Sqrt[(a + Cos[\[Theta]])/(
 1 + a)], Assumptions->a>1]

or try this:
    Simplify[ Sqrt[a + Cos[\[Theta]]] /Sqrt[(a + Cos[\[Theta]])/(
 1 + a)], Assumptions->a>1]

