Calculate ∫0π/2(sin2(sinx)+cos2(cosx))dx.
Prove that for x≠πn (n is an integer) sinx and cosx are rational if and only if the number tanx/2 is rational.
Does there exist a real number α such that the number cosα is irrational, and all the numbers cos2α, cos3α, cos4α, cos5α are rational?