statements about summation

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Could you help me prove this statements about summation?

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I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why?

Do you know how can I put explicitly that sumation of cossine and sine according the first and the second members. I were thinking it, because I'd calcule the area of a sine function in a given distance...

Thank you.