Solving the equation \(\displaystyle{\tan{{\left\lbrace{x}\right\rbrace}}}+{\tan{{\frac{{{x}}}{{{4}}}}}}={2}\) 1st attempt: I use

Oliver Carson

Oliver Carson

Answered question

2022-03-31

Solving the equation tan{x}+tanx4=2
1st attempt: I use the following identity tan2x=2tanx1tan2x. Then get the following equation
u52u48u3+12u2+3u+2=0
where u=tanx4. But i can't find any root for this equation.
2nd attempt:
tan{4x}+tanx=2sin{4x}cos{x}+sin{x}cos{4x}=2cos{4x}cos{x}sin{5x}=2cos{4x}cos{x}
sin{5x}=2cos{4x}cos{x}12sin{5x}=cos{4x}cos{x}cos({2kπ±π3})sin{5x}=cos{4x}cos{x}
I can't continue from here.

Answer & Explanation

pypberissootcu

pypberissootcu

Beginner2022-04-01Added 14 answers

you can use
tan(x)=2tan(x2)1tan2(x2)
and
tan(x2)=2tan(x4)1tan2(x4)
putting all things together and factorizing we get
({u}24,u+1)({u}3+2,{u}23,u2)=0
where
u=tan(x4)

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