$$\frac{1}{3+i}-\frac{1}{3-i}$$
$-\frac{1}{5}i=-0.2i$
$$\frac{1\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}-\frac{1}{3-i}$$
$$\frac{3-i}{10}-\frac{1}{3-i}$$
$$\frac{3}{10}-\frac{1}{10}i-\frac{1}{3-i}$$
$$\frac{3}{10}-\frac{1}{10}i-\frac{1\left(3+i\right)}{\left(3-i\right)\left(3+i\right)}$$
$$\frac{3}{10}-\frac{1}{10}i-\frac{3+i}{10}$$
$$\frac{3}{10}-\frac{1}{10}i+\left(-\frac{3}{10}-\frac{1}{10}i\right)$$
$$-\frac{1}{5}i$$
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$0$
$$Re(\frac{1\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}-\frac{1}{3-i})$$
$$Re(\frac{3-i}{10}-\frac{1}{3-i})$$
$$Re(\frac{3}{10}-\frac{1}{10}i-\frac{1}{3-i})$$
$$Re(\frac{3}{10}-\frac{1}{10}i-\frac{1\left(3+i\right)}{\left(3-i\right)\left(3+i\right)})$$
$$Re(\frac{3}{10}-\frac{1}{10}i-\frac{3+i}{10})$$
$$Re(\frac{3}{10}-\frac{1}{10}i+\left(-\frac{3}{10}-\frac{1}{10}i\right))$$
$$Re(-\frac{1}{5}i)$$
$$0$$
