The second question is proved by induction, because when n = 1, cosnA is a rational number, right?
So let's assume that coskx is a rational number. Now let's prove that cos(k+ 1)x is also a rational number.
According to the sum angle formula, it looks simple. . .
Cos (k+1) x = cos (kx+x) = coskxcosx-sinksinx are all rational numbers, because the four operation results of rational numbers are still rational numbers, which is the basic theorem of rational numbers.