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  4. If log π x>0 then the absolute value of log (1/π)( sin -1 (2 x/1+x2)+2 tan -1 x) is equal to

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Q. If $\log _{\pi} x>0$ then the absolute value of $\log _{\frac{1}{\pi}}\left(\sin ^{-1} \frac{2 x}{1+x^{2}}+2 \tan ^{-1} x\right)$ is equal to

Inverse Trigonometric Functions

Solution:

Since $\log _{\pi} x > 0 \Rightarrow x>1$
For $x > 1, \sin ^{-1} \frac{2 x}{1+x^{2}}=\pi-2 \tan ^{-1} x$
$\log _{\frac{1}{\pi}}\left(\pi-2 \tan ^{-1} x+2 \tan ^{-1} x\right)$
$=\log _{\frac{1}{\pi}}(\pi)=-1$