sinx+sin2x+sin3x+sin4x+sin5x+sin6x=0
\[\left(sin6x+sinx \right)+\left(sin4x+sin3x \right)+\left(sin5x+sin2x \right)=0\]
\[\Leftrightarrow 2sin\left(\frac{7x}{2} \right).cos\left(\frac{5x}{2} \right)+2sin\left(\frac{7x}{2} \right).cos\left(\frac{3x}{2} \right)+2sin\left(\frac{7x}{2} \right).cos\left(\frac{x}{2} \right)=0\]
\[\Leftrightarrow sin\left(\frac{7x}{2} \right).[cos\left(\frac{5x}{2} \right) \right+cos\left(\frac{3x}{2} \right)+cos\left(\frac{x}{2}\right) \right)]=0\]
\[\Leftrightarrow sin\left(\frac{7x}{2} \right).[cos\left(\frac{3x}{2} \right)+2cos\left(\frac{3x}{2} \right).cox\left(x \right) =0]\]
\[\Leftrightarrow sin\left(\frac{7x}{2} \right).cos\left(\frac{3x}{2} \right).\left(1+2cosx \right)=0\]
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