Can You Prove the Convergence of a Trigonometric Series?

[Mr. Rho]

Homework Statement

I need to show that \sum\limits_{n=0}^\infty \frac{sin^{4}(\frac{n\pi}{4})}{n^2} = \frac{\pi^{2}}{16}

Homework Equations

I have this property for odd n

\sum\limits_{n=0}^\infty \frac{1}{n^2} = \frac{\pi^{2}}{8}

The Attempt at a Solution

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I have no idea how to do this, any help?

 

[haruspex]

There are only so many values that $\sin(\frac{n\pi}{4})$ can take. How about splitting the sum up on that basis?