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@:Sinx+Sin2x+Sin3x+Sin4x+Sin5x+Sin6x=0
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[TEX](sinx + sin6x) + ( sin2x + sin5x) + ( sin3x + sin4x) = 0[/TEX]
[TEX]2sin{\frac{7x}2}cos{\frac{5x}2 }+ 2sin{\frac{7x}2}cos{\frac{3x}2} + 2sin{\frac{7x}2}cosx = 0[/TEX]
[TEX]2sin{\frac{7x}2} ( cos{\frac{5x}2 } + cos{\frac{3x}2} + cosx) = 0[/TEX]
[TEX]2sin{\frac{7x}2}.[(cos{\frac{x}2} + cos{\frac{3x}2}) + cosx] =0[/TEX]
[TEX] 2sin{\frac{7x}2} . [ 2cosxcos{\frac{x}2} + cosx] = 0 [/TEX]
[TEX]2sin{\frac{7x}2}. cosx ( 2cos{\frac{x}2} + 1) = 0 [/TEX]
\Rightarrow [TEX]\left[sin{\frac{7x}2} = 0 \\ cosx = 0 \\ cos{\frac{x}2} = \frac{-1}2 [/TEX]