1.

For each natural number `nge2`, prove that `sinx_1cosx_2+sinx_2cosx_3+…+sinx_ncosx_1len//2`(where `x_1,x_2,…,x_n` are arbitrary real numbers).

Answer» Let the required sum be `S_(n)`. We know that
`(sinx_1-cosx_2)^2=(sinx_2-cosx_3)^2+...+(sinx_(n-1)-cosx_n)^2+(sinx_n-cosx_1)^2ge0`
or `(sin^2x_1-cos^2x_2)+(sin^2x_2-cos^2x_3)(sin^2x_3+cos^2x_3)+...+(sin^2x_(n)-cos^2x_n)ge2S_n`
`rArr nge2S_n`
or `S_nlen//2`


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