This is a question in introduction to pure mathematics. I am pretty sure I am close to the answer but I can't quite decide why this proves that there is no complex numbers:

$$|z| = |z + i√5| = 1$$
$$\sqrt{cos²Θ + (isinΘ + i\sqrt5)²} = \sqrt1$$
$$cos²Θ - sin²Θ - 2\sqrt5sinΘ - 5 = 1$$
$$cos²Θ - (1 - cos²Θ) - 2\sqrt5sinΘ - 5 = 1$$
$$2cos²Θ - 2√5sinΘ - 5 = 0$$

I can see are that there is no imaginary component left in the equation and there is no way to equate it to $cos\theta + isin\theta$.