I think this might be right if you look at the parameters for which 1 - (x^2)/2 is lower than cos(x). i.e. What looks like between the parameters of -2 <= x <= -0.5 and .5 <= x <= 2. That solves the inequality on the left of 1 - (x^2)/2 <= cos(x). Cos x <= 1 is the true for all of x. Therefore for them to both be true you use the answer for the first one. The [-2,2] comes from the notion for both to hold, we cos(x) cannot be lower than -1, therefore at most 1 - (x^2)/2 = -1, allows for x to be anywhere in the [-2,2] domain for the range to remain [-1,1]. The limit for part two is 1, that's clear from the graph but mostly from entering the limit.
