Solve for x: ^ -1 ( x - 1 x - 2 )+ ^ -1 ( x + 1 x + 2 )= 4 . - Vidyadip

Question

Solve for x:$\tan^{-1}\Bigg(\frac{\text{x - 1}}{\text{x - 2}}\Bigg)+\tan^{-1}\Bigg(\frac{\text{x + 1}}{\text{x + 2}}\Bigg)=\frac{\pi}{4}.$

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Answer

$\tan^{-1}\Bigg(\frac{\text{x - 1}}{\text{x - 2}}\Bigg)+\tan^{-1}\Bigg(\frac{\text{x + 1}}{\text{x + 2}}\Bigg)=\frac{\pi}{4}$
$\tan^{-1}\Bigg(\frac{\frac{\text{x - 1}}{\text{x - 2}}+\frac{\text{x + 1}}{\text{x + 2}}}{1-\frac{\text{x - 1}}{\text{x - 2}}\times\frac{\text{x + 1}}{\text{x + 2}}}\Bigg)=\frac{\pi}{4}$
$\tan^{-1}\Bigg(\frac{\text{2x}^{2}-4}{-3}\Bigg)=\frac{\pi}{4}.$
$\Rightarrow\frac{\text{2x}^{2}-4}{-3}$ = 1
$\Rightarrow\text{2x}^{2}$ = 1
$\Rightarrow\text{x}$ $=\pm\frac{1}{\sqrt{2}}$.

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