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If $x\, sin \theta = y\, sin \, \left( {\theta \,\, + \,\,\frac{{2\,\pi }}{3}} \right) = z\, sin \, \left( {\theta \,\, + \,\,\frac{{4\,\pi }}{3}} \right)$ then :
$x + y + z = 0$
$xy + yz + zx = 0$
$xyz + x + y + z = 1$
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Solution
$\frac{x}{y}\,\, = \,\frac{{\sin 2\pi /3\,.\,\,\cos \theta \,\, + \,\cos 2\pi /3\,.\sin \theta }}{{\sin \theta }}$
$=$ $\frac{1}{2}\,\left[ {\frac{{\sqrt 3 \,\cos \theta \, – \,\sin \theta }}{{\sin \theta }}} \right]$
$=$$\frac{{\sqrt 3 }}{2}\,\cot \theta \, – \,\frac{1}{2}$ ….$(1)$
$|||^{1y}$ $\frac{x}{z}\,\, = \,\,\frac{{\sin \theta \,.\,\cos 4\pi /3\,\, + \,\,\cos \theta \,.\,\sin 4\pi /3}}{{\sin \theta }}$
$=$ $ – \,\,\frac{1}{2}\,\, – \,\,\frac{{\sqrt 3 }}{2}\,\,\cot \theta $ ….$(2)$
$\frac{x}{y}\,\, + \,\,\frac{x}{z}\,\, = \,\, – 1$
$\Rightarrow xz + xy + yz = 0$
