Proof of complex number identity

[Gameowner]

Homework Statement

Attached question

Homework Equations

The Attempt at a Solution

The second part of question is relatively easy, it is the first part of the question where I need help with(using arg zw = arg z + arg w to show arg z^n = n arg z).

Also, is the question asking to proof de Moivre's formula with the identity arg zw = arg z + arg w? or just proof LHS=RHS?

 

[alxm]

I guess it's the induction that's causing problems. Show that it's true for z². Then show that if it's true for zⁿ, that it's also true for zⁿ⁺¹.

The second part uses the relation you proved in the first part.

 

[eumyang]

The base case would be for z¹, I thought. In any event, it's easy to prove.
The inductive case: assume true for n = k, show that it's true for n = k + 1.
Here's a hint: x^{a + 1}= (x^a)(x).69