<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] Expand the equation:
$$
\begin{align}
\sqrt{\left(\frac{a}{a^2+b^2}\right)^2+\left(\frac{b}{a^2+b^2}\right)^2}&=\sqrt{\left(\frac{a}{c^2}\right)^2+\left(\frac{b}{c^2}\right)^2}\\
&=\sqrt{\frac{a^2}{c^4}+\frac{b^2}{c^4}}\\
&=\sqrt{\frac{c^2}{c^4}}\\
&=\sqrt{\frac{1}{c^2}}\\
&=\frac{1}{c}
\end{align}
$$
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