Let's invoke De Moivre's theorem here. We have
Since is a positive integer, the binomial theorem-N choose K holds for .
Hence, by expanding, we have
Since we have
By equating the real and imaginary parts, we obtain
where the terms of both the above series are alternately positive and negative. Also, each series continues till one of the factors in the numerator is zero and then ceases. Hence, proved.
To prove for , observe that
Divide the numerator and the denominator by to obtain
Hence, our proof is complete.