The number of integral values of x satisfying the equation ${tan}^{- 1} (3 x) + {tan}^{- 1} (5 x) = {tan}^{- 1} (7 x) + {tan}^{- 1} (2 x)$ is

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Solution

We know, ${tan}^{- 1} x + {tan}^{- 1} y = {tan}^{- 1} (\frac{x + y}{1 - x y})$
Applying the above expression, the above question simplifies to
$\frac{5 x + 3 x}{1 - 15 x^{2}} = \frac{7 x + 2 x}{1 - 14 x^{2}}$
$8 x (1 - 14 x^{2}) = 9 x (1 - 15 x^{2})$
$8 x - 112 x^{3} = 9 x - 135 x^{3}$
$23 x^{3} - x = 0$
$x (23 x^{2} - 1) = 0$
$x = 0$ and $x = \pm \frac{1}{\sqrt{23}}$
Hence only one integral solution at $x = 0$ .

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