January 3rd, 2017,
Since this question involves integral calculus, I have moved it here.
In order for two functions to have the same anti-derivative, they must in fact be equivalent. From a double-angle identity for cosine, we know:
$ \displaystyle \cos(2x)=1-2\sin^2(x)$
$ \displaystyle -\cos(2x)=2\sin^2(x)-1\ne2\sin^2(x)$
Thus, the two given functions are not equivalent, therefore they cannot have the same anti-derivative.
Can you use a similar line of reasoning for the second question?