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Introductory complex number identity mistake?

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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)).
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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.

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