<s>[INST] How do I derive the formula for the reciprocal of a hypotenuse? Given $a^2 + b^2 = c^2$,
Why is it that the equation below yields the reciprocal of the hypotenuse c, ($\frac{1}c$)?
$\sqrt{(\frac{a}{a^2+b^2})^2 + (\frac{b}{a^2+b^2})^2}$ 

Worked example: 
 $3^2 + 4^2 = c^2$
 $c = 5$
 $\sqrt{(\frac{3}{3^2+4^2})^2 + (\frac{4}{3^2+4^2})^2}$
 = $\sqrt{(\frac{3}{25})^2 + (\frac{4}{25})^2}$

= $\sqrt{(.0144) + (.0256)}$
 = $\sqrt{.04}$
 = .2 or $1/5$
 [/INST] Observe that, for $ab\ne0$,
$$
\left(\frac{a}{a^2+b^2}\right)^2 + \left(\frac{b}{a^2+b^2}\right)^2=\frac{a^2}{(a^2+b^2)^2}+\frac{b^2}{(a^2+b^2)^2}=\frac{a^2+b^2}{(a^2+b^2)^2}=\frac{1}{a^2+b^2}.
$$ Hope you can take it from here.
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