How can I get Mathematica to simplify $\Vert\{1,\sin(t),\cos(t)\}\Vert_2$ to $\sqrt{2}$?

Question

Context

Norm[{1, Sin[t], Cos[t]}]
Norm[{1, Sin[t], Cos[t]}] // Simplify
(*
Sqrt[1 + Abs[Cos[t]]^2 + Abs[Sin[t]]^2]    
Sqrt[1 + Abs[Cos[t]]^2 + Abs[Sin[t]]^2]
*)

Question:

How do I coax Mathematica to output Sqrt[2]?

Answer 1

Simplify[Norm[{1, Sin[t], Cos[t]}], Element[t, Reals]]

Because

Plot3D[Norm[{1, Sin[a + b I], Cos[a + b I]}], {a, -1, 1}, {b, -1, 1}]

Answer 2

Just so that all the i's are dotted and t's crossed:

$$\cos^2(z)+\sin^2(z)=1$$ for all complex $z$ (since $\cos^2(z)+\sin^2(z)-1$ is a holomorphic function that vanishes on the real axis, it vanishes everywhere); but $$|\cos(z)|^2+|\sin(z)|^2 \neq 1$$ in general unless $z$ is real.