Simplify a trigonometric expession with Mathematica

Question

I would need to simplify or approximate this expression:

Abs[0.00178442 Cos[alpha] - 0.0368414 Sin[alpha] + (0.998216 Cos[alpha] + 0.0368414 Sin[alpha]) (Cos[2 dx] + I Sin[2dx])]^4

I have tried with the Expand and Simplify/FullSimplify commands in Mathematica but it does not calculate the Abs[]^4. Could anyone help with me with this issue?

Answer 1

Instead of taking (Abs[x])^4, you can work with Abs[x^4]:

x=0.00178442 Cos[alpha] - 0.0368414 Sin[alpha] + (0.998216 Cos[alpha] + 0.0368414 Sin[alpha]) (Cos[2 dx] + I Sin[2dx]);
Abs[Expand[x^4]]//FullSimplify
(*Justification: verifying that Abs[x]^4 = Abs[x^4]*)
Resolve[ForAll[z, Abs[z]^4 == Abs[z^4]]]

It still looks like a messy expression, but maybe you can make progress from here