Deriving the Relationship of Cubic Crystal Structure

[alexgmcm]

How can one derive the relationship for a cubic crystal:
d= \frac{a}{\sqrt{h^2 + k^2 + l^2}}

It is shown here.

This is probably trivial but I am having trouble proving it as the Miller indices are reciprocals but the reciprocal is not distributive. This isn't for homework by the way - I am just going over solid state stuff from last year trying to get a more solid understanding than I got when I just learned it by focusing on the then impending exams.

Any help would be greatly appreciated. :)

 

[alexgmcm]

I found the solution had already been posted here: https://www.physicsforums.com/showthread.php?t=119037

Can someone explain why the sum of the squares of the cosines is equal to one though? It just says 'from trig' but I'm not familiar with that identity :/