Hey Gilbert.

Hint: Try going from x^(2n+2)/(2n+2)! * (-1)^n to x^(2n)/(2n)!*(-1)^n by changing the limits n=0 to infinity to n=1 to infinity and then use the fact that:

cos(x) = 1 - Sigma(n=1) x^(2n)/(2n)! * (-1)^n which implies

Sigma(n=1) x^(2n)/(2n)! * (-1)^n = 1 - cos(x)