complex analysis

On the assumption that alpha, beta, a and b are real,

$$\alpha -\imath \beta=\frac{1}{a-\imath b}=\frac{a+\imath b}{a^{2}+b^{2}}.$$

Taking the complex conjugate of both sides,

$$\alpha + \imath\beta=\frac{a-\imath b}{a^{2}+b^{2}},$$

and multiplying the equations together,

$$\alpha^{2}+\beta^{2}=\frac{a^{2}+b^{2}}{(a^{2}+b^{2})^{2}},$$

from which the result follows.

I think you either forgot to mention some of the details, or the variables (other than i off course) have an implicit meaning which I'm unaware of. 

Anyway, I can make it a little easier for you by doing the following. 

$$\begin{array}{lcl} \alpha - i\beta &= \frac{1}{a-ib}\\ \alpha - i\beta &= \frac{1}{a-ib}\frac{a+ib}{a+ib}\\ \alpha - i\beta &= \frac{a+ib}{a^2+ aib - aib -i^2b^2}\\ \alpha - i\beta &= \frac{a+ib}{a^2+b^2}\\ (\alpha - i\beta)(\alpha + i\beta) &= \frac{(a+ib)(\alpha + i\beta)}{a^2+b^2}\\ \alpha^2 +i\alpha \beta - i\alpha \beta - i^2 \beta^2 &= \frac{(a+ib)(\alpha + i\beta)}{a^2+b^2}\\ \alpha^2 + \beta^2 &= \frac{(a+ib)(\alpha + i\beta)}{a^2+b^2}\\ (\alpha^2 + \beta^2)(a^2+b^2)&= (a+ib)(\alpha + i\beta) \end{array}$$

 Now if you can prove that

$$(a+ib)(\alpha + i\beta) = 1$$

You're there.

Reinout