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Question Number 198562 by cortano12 last updated on 22/Oct/23

$$\mathrm{f}\left(\tan \mathrm{x}\right)+2\mathrm{f}\left(\cot \mathrm{x}\right)=4\mathrm{x}$$$$\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=?$$

Answered by dimentri last updated on 22/Oct/23

$$letx=\frac{\pi }{2}-t$$$$f\left(\cot t\right)+2f\left(\tan t\right)=2\pi -4t$$$$\{\begin{array}{l}f\left(\cot x\right)+2f\left(\tan x\right)=2\pi -4x\\ 2f\left(\cot t\right)+f\left(\tan x\right)=4x\end{array}$$$$\{\begin{array}{l}2f\left(\cot x\right)+4f\left(\tan x\right)=4\pi -8x\\ 2f\left(\cot x\right)+f\left(\tan x\right)=4x\end{array}$$$$\iff f\left(\tan x\right)=\frac{4\pi -12x}{3}$$$$\iff f\left(x\right)=\frac{4\pi -12{\tan }^{-1}x}{3}$$

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