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Question
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
Solution
\[\text{ Given }: \]
\[x \cos\theta = y\left( \cos\theta\cos\frac{2\pi}{3} - \sin\theta \sin\frac{2\pi}{3} \right) = z\left( \cos\theta\cos\frac{4\pi}{3} - \sin\theta \sin\frac{4\pi}{3} \right)\]
\[ \Rightarrow x\cos\theta = y\left( - \frac{1}{2}\cos\theta - \frac{\sqrt{3}}{2}\sin\theta \right) = z\left( - \frac{1}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta \right) \]
\[ \Rightarrow x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right) = \frac{z}{2}\left( - 1 + \sqrt{3}\tan\theta \right)\]
\[x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right)\]
\[z = \frac{y\left( - 1 - \sqrt{3}\tan\theta \right)}{\left( - 1 + \sqrt{3}\tan\theta \right)}\]
\[\text{ Now }, \]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{y\left( - 1 - \sqrt{3}\tan\theta \right)} + \frac{1}{y} + \frac{\left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = \frac{2 + \left( - 1 - \sqrt{3}\tan\theta \right) + \left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = 0\]
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