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Q)

If $3\tan^{-1}x+4\cot^{-1}x=2\pi$ then $ x$ =?

$(a)\;1\qquad(b)\;0\qquad(c)\;-1\qquad(d)\;\large\frac{1}{\sqrt{3}}$

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A)
Toolbox:
  • \( tan^{-1}x+cot^{-1}x=\large\frac{\pi}{2} \)
  • \(tan^{-1}1=\large\frac{\pi}{4}\)
Ans (B) 1
Given: \(3tan^{-1}x+4cot^{-1}x=2\pi \)
We can write the given equation as
\(3tan^{-1}x+3cot^{-1}x+cot^{-1}x=2\pi \)
\( \Rightarrow\:3(tan^{-1}x+cot^{-1}x)+cot^{-1}x=2\pi\)
\( \Rightarrow 3(\large\frac{\pi}{2})\)\(+cot^{-1}x=2\pi \)
\( \Rightarrow cot^{-1}x=2\pi-3\large\frac{\pi}{2}=\frac{\pi}{2}\)
\(\Rightarrow x = 0\)
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