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For each natural number nlt=2, prove tha...

For each natural number `nlt=2,` prove that `sinx_1cosx_2+sinx_2c0sx+3++sinx_ncosx_1lt=n/2` (where `x_1, x_2, ,x_n` are arbitrary real numbers).

Text Solution

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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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