Question

Introductory complex number identity mistake?

Answer 1

For complex numbers, the 'absolute value' bars || are used to represent the modulus of the number.  The modulus of a complex number, which I'll call M, is given by M = √(z · z\*), where z is a general complex number and z\* is its complex conjugate.

Given that a complex conjugate of a complex number (a ± bi) is (a ∓ bi), you should be able to see that your modulus here works to √(cos^(2)(φ) - i^(2)sin^(2)(φ)), or √(cos^(2)(φ) + sin^(2)(φ)).

Answer 2

I think I found my mistake. When the absolute value of any number is taken, including complex, the minus sign is omitted and replaced by a positive sign.