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I've stumbled on the equality $$ \tan ^{-1}\left(\frac{3}{4}\right) \left(\pi -\tan ^{-1}\left(4 \sqrt{3}\right)\right)=4 \tan ^{-1}\left(\frac{2}{\sqrt{3}}\right) \cot ^{-1}(3). $$ Out of curiosity, is there a simple proof? I don't see how it can be done with standard formulas like $\cot^{-1}x=\pi/2-\tan^{-1}x$.

Andrew
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HINT:

For $x>0,\cot^{-1}x=\tan^{-1}1/x$

Using the formula in my answer here

$2\tan^{-1}(1/3)=\tan^{-1}\dfrac{2\cdot1/3}{1-(1/3)^2}=\tan^{-1}3/4$

and use the same formula for $2\tan^{-1}\dfrac2{\sqrt3}$

Observe that $\dfrac2{\sqrt3}>1$

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lab bhattacharjee
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