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f-tan-x-2f-cot-x-4x-f-x-




Question Number 198562 by cortano12 last updated on 22/Oct/23
   f(tan x)+ 2f(cot x) = 4x      f ′(x)= ?
$$\:\:\:\mathrm{f}\left(\mathrm{tan}\:\mathrm{x}\right)+\:\mathrm{2f}\left(\mathrm{cot}\:\mathrm{x}\right)\:=\:\mathrm{4x}\: \\ $$$$\:\:\:\mathrm{f}\:'\left(\mathrm{x}\right)=\:? \\ $$
Answered by dimentri last updated on 22/Oct/23
 let x=(π/2)−t    f(cot t) +2 f(tan t)= 2π−4t     { ((f(cot x)+2 f(tan x)=2π−4x)),((2 f(cot t)+f(tan x)= 4x)) :}     { ((2f(cot x)+4f(tan x)=4π−8x)),((2f(cot x)+f(tan x)= 4x)) :}   ⇔ f(tan x)=((4π−12x)/3)   ⇔ f(x)= ((4π−12 tan^(−1) x)/3)
$$\:{let}\:{x}=\frac{\pi}{\mathrm{2}}−{t}\: \\ $$$$\:{f}\left(\mathrm{cot}\:{t}\right)\:+\mathrm{2}\:{f}\left(\mathrm{tan}\:{t}\right)=\:\mathrm{2}\pi−\mathrm{4}{t} \\ $$$$\:\:\begin{cases}{{f}\left(\mathrm{cot}\:{x}\right)+\mathrm{2}\:{f}\left(\mathrm{tan}\:{x}\right)=\mathrm{2}\pi−\mathrm{4}{x}}\\{\mathrm{2}\:{f}\left(\mathrm{cot}\:{t}\right)+{f}\left(\mathrm{tan}\:{x}\right)=\:\mathrm{4}{x}}\end{cases} \\ $$$$\:\:\begin{cases}{\mathrm{2}{f}\left(\mathrm{cot}\:{x}\right)+\mathrm{4}{f}\left(\mathrm{tan}\:{x}\right)=\mathrm{4}\pi−\mathrm{8}{x}}\\{\mathrm{2}{f}\left(\mathrm{cot}\:{x}\right)+{f}\left(\mathrm{tan}\:{x}\right)=\:\mathrm{4}{x}}\end{cases} \\ $$$$\:\Leftrightarrow\:{f}\left(\mathrm{tan}\:{x}\right)=\frac{\mathrm{4}\pi−\mathrm{12}{x}}{\mathrm{3}} \\ $$$$\:\Leftrightarrow\:{f}\left({x}\right)=\:\frac{\mathrm{4}\pi−\mathrm{12}\:\mathrm{tan}^{−\mathrm{1}} {x}}{\mathrm{3}} \\ $$

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