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  4. The number of solution of the equation 2 tan -1 x+ cot -1 x=(7 π/6) is

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Q. The number of solution of the equation $2 \tan ^{-1} x+\cot ^{-1} x=\frac{7 \pi}{6}$ is

Inverse Trigonometric Functions

Solution:

$ 2 \tan ^{-1} x+\cot ^{-1} x=\frac{7 \pi}{6} \Rightarrow \tan ^{-1} x+\frac{\pi}{2}=\frac{7 \pi}{6} \Rightarrow \tan ^{-1} x=\frac{2 \pi}{3} $
$\Rightarrow \tan ^{-1} x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)(x \in \phi)$