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.

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:
$$\frac{a^2+b^2}{c^2+d^2}=x^2+y^2$$
