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Rational cosin and sin

Source: All Russian Olympiad 2017,Day1,grade 11,P1

May 1, 2017

number theorytrigonometry

Problem Statement

S=\sin{64x}+\sin{65x}

and

C=\cos{64x}+\cos{65x}

are both rational for some

x

. Prove, that for one of these sums both summands are rational too.