Pessimist Singularitarian

#2
January 3rd, 2017,
11:11
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)$

Hence:

$ \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?