Question:
$5 \tan ^{-1} x+3 \cot ^{-1} x=2 \pi$
Solution:
$5 \tan ^{-1} x+3 \cot ^{-1} x=2 \pi$
$\Rightarrow 5 \tan ^{-1} x+3\left(\frac{\pi}{2}-\tan ^{-1} x\right)=2 \pi$ $\left[\because \cot ^{-1} x=\frac{\pi}{2}-\tan ^{-1} x\right]$
$\Rightarrow 5 \tan ^{-1} x+\frac{3 \pi}{2}-3 \tan ^{-1} x=2 \pi$
$\Rightarrow 2 \tan ^{-1} x=\frac{\pi}{2}$
$\Rightarrow \tan ^{-1} x=\frac{\pi}{4}$
$\Rightarrow x=\tan \frac{\pi}{4}=1$
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