(1). In the case of fractions, the domain is restricted by the denominator. The function is undefined when the denominator is zero. In this case 1-e1-x²=0 when e1-x²=1, that is, when 1-x2=0=(1-x)(1+x).
At x=1 or x=-1 the function is undefined so the domain is {(-∞,-1), (-1,1), (1,∞)}.
(2) ecos(x)=0 can never happen, so the domain is unrestricted: (-∞,∞).