Computing symbolic norm of a vector

Question

Consider the code:

n1 := Cos[α] Sin[β]
n2 := Sin[α] Sin[β]
n3 := Cos[β]
n := {n1, n2, n3}
Norm[n]

Why the result is

Sqrt[Abs[Cos[β]]^2 + Abs[Cos[α] Sin[β]]^2 + Abs[Sin[α] Sin[β]]^2]

that is $$\sqrt{\left| \sin (\alpha ) \sin (\beta )\right| ^2+\left| \cos (\alpha ) \sin (\beta )\right| ^2+\left| \cos (\beta )\right| ^2}$$

instead of $1$?

How to compute the norm in a symbolic way simplifying some trigonometric identities?

Answer 1

n1 := Cos[α] Sin[β]
n2 := Sin[α] Sin[β]
n3 := Cos[β]
n := {n1, n2, n3}

FullSimplify[Norm[n], {α, β} ∈ Reals]

(* 1 *)

Answer 2

For real values, there is another way, because then you don't need the Abs which is created by Norm. It is simply

Simplify[Sqrt[n.n]]

(* 1 *)