[Probability] Confusing change of a random variable

Can you work out the probability that X is even/odd?
The probability (-1)^(X) = 1 is the sum of the probabilities X = k over k = 0,2,4,6,8,... and for (-1)^(X) = -1 over k = 1,3,5,7,... Hint: relate the sums to the power series for cosh(x) and sinh(x).

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