How is sin(npi/2) = (-1)^((n-1)/2) for odd n?

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I know the MacLaurin series for sin(x) but can't figure out how this reduces to the above (if it does).
 

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sin(npi/2) is going to be equal to -1 or 1, depending on the value of n, right? So the formula seems plausible. You could prove it through induction.
 
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