<s>[INST] Taking Mod on both sides, mathematically correct? When given a equation containing complex numbers such as 
$$  \frac{a+ib}{c+id} = x + iy$$
and required to prove 
$$ \frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$
Is taking the mod of both sides a legal mathematical step? I ask so because my textbook first finds the conjugate of both sides and multiplies the conjugate with the initial equation. This is done everywhere a case like the above one occurs. Hence im a bit confused if the way im thinking of proceeding is mathematically legal.
 [/INST] Yes, taking the mod of both sides is mathematically valid, but we don't necessarily need to do that.

Convert to polar:
$$\frac{\sqrt{a^2+b^2}\text{cis}(\theta_1)}{\sqrt{c^2+d^2}\text{cis}(\theta_2)}=\sqrt{x^2+y^2}\text{cis}(\theta_3)$$
Separate the moduli and arguments:
$$\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}\sqrt{x^2+y^2}}=\frac{\text{cis}(\theta_3)\text{cis}(\theta_2)}{\text{cis}(\theta_1)}=\text{cis}(\theta_3+\theta_2-\theta_1)$$
The left-side is obviously a positive real number, but the only positive real number in the range of $\text{cis}$ is $1$, so we have:
$$\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}\sqrt{x^2+y^2}}=1$$
Multiply both sides by $\sqrt{x^2+y^2}$:
$$\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}=\sqrt{x^2+y^2}$$
Square both sides:
$$\frac{a^2+b^2}{c^2+d^2}=x^2+y^2$$
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