Probability that x is even if follows poisson distribution

Question

Assume that X follows Poisson distribution, POISSON. Taylor expansion for exponent (e^t = 1 + t/1! + ... + t/n! + ...) is valid for any real t. Combine expansions for (e^t) and for (e^(-t)) to determine probabilities below.

i) Determine probability that X is even (that is 0, 2 ... 2n).

ii) Determine probability that X is odd.