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If $\mathrm{x}, \mathrm{x}-\mathrm{y}$ and $\mathrm{x}+\mathrm{y}$ are the angles of a triangle (not an equilateral triangle) such that $\tan (\mathrm{x}-\mathrm{y}), \tan \mathrm{x}$ and $\tan (\mathrm{x}+\mathrm{y})$
are in GP, then what is $x$ equal to?
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are in GP, then what is $x$ equal to?
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Verified Answer
The correct answer is:
$\frac{\pi}{3}$
$\mathrm{x}, \mathrm{x}-\mathrm{y}, \mathrm{x}+\mathrm{y}$ are angles of a triangle. $\tan (\mathrm{x}-\mathrm{y}), \tan \mathrm{x}, \tan (\mathrm{x}+\mathrm{y})$ are in G.P.
Now, $\mathrm{x}+\mathrm{x}-\mathrm{y}+\mathrm{x}+\mathrm{y}=\pi$ (Sum of angles in triangle $\left.=180^{\circ}=\pi\right)$
$\Rightarrow 3 x=\pi$
$\Rightarrow x=\frac{\pi}{3}$
Now, $\mathrm{x}+\mathrm{x}-\mathrm{y}+\mathrm{x}+\mathrm{y}=\pi$ (Sum of angles in triangle $\left.=180^{\circ}=\pi\right)$
$\Rightarrow 3 x=\pi$
$\Rightarrow x=\frac{\pi}{3}$
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