Simplifying Ugly Radical Expressions from Cardano's Method

I asked a similar question to this some time ago, but as we have gained several new specialists since then, I thought I'd try again.

The problem comes from solving the equation

$\displaystyle x^3 - 7x + 6 = 0$

by Cardano's method.

I get an expression for x of the form:

$\displaystyle x = 2\sqrt[3]{\frac{7 \sqrt{21}}{9}}cos \left ( \frac{1}{3} atn \left ( \frac{10\sqrt{3}}{27} \right ) \right )$

As it happens,

$\displaystyle 2\sqrt[3]{\frac{7 \sqrt{21}}{9}}cos \left ( \frac{1}{3} atn \left ( \frac{10\sqrt{3}}{27} \right ) \right ) = 3$

but how can I prove that? Is there any way to simplify this monster without knowing the answer ahead of time?

-Dan