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Question Number 130589 by bramlexs22 last updated on 27/Jan/21

$$\mathrm{Prove}\mathrm{the}\mathrm{identity}{\tan }^{-1}\left(\mathrm{x}\right)+{\cot }^{-1}\left(\mathrm{x}\right)=\pi /2$$

Answered by EDWIN88 last updated on 27/Jan/21

$$Iff\left(x\right)={\tan }^{-1}\left(x\right)+{\cot }^{-1}\left(x\right)$$$$thenf{}^{\prime }\left(x\right)=\frac{1}{1+x^2}-\frac{1}{1+x^2}=0$$$$for\mathrm{\forall }valuesofx.Therefore$$$$f\left(x\right)=C,aconstant.$$$$TodeterminethevalueofC$$$$weputx=1sincewecanevaluate$$$$f\left(1\right)exactly.ThenC=f\left(1\right)=$$$${\tan }^{-1}\left(1\right)+{\cot }^{-1}\left(1\right)=\frac{\pi }{4}+\frac{\pi }{4}=\frac{\pi }{2}$$

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