Question Number 201460 by hardmath last updated on 06/Dec/23
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{period}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{function}: \\ $$$$\mathrm{y}\:=\:\mid\:\mathrm{tan}\:\mathrm{2x}\:\mid\:\:+\:\:\mid\:\mathrm{cot}\:\mathrm{2x}\:\mid \\ $$
Answered by Mathspace last updated on 07/Dec/23
$${y}\left({x}\right)=\mid{tan}\left(\mathrm{2}{x}\right)\mid+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}{x}\right)\mid} \\ $$$${y}\left({x}+\frac{\pi}{\mathrm{2}}\right)=\mid{tan}\left(\mathrm{2}\left({x}+\frac{\pi}{\mathrm{2}}\right)\right)+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}\left({x}+\frac{\pi}{\mathrm{2}}\right)\right)} \\ $$$$=\mid{tan}\left(\mathrm{2}{x}+\pi\right)\mid+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}{x}+\pi\right)\mid} \\ $$$$=\mid{tan}\left(\mathrm{2}{x}\right)\mid+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}{x}\right)\mid} \\ $$$$={f}\left({x}\right)\:{so}\:{the}\:{small}\:{period}\:{is} \\ $$$${T}=\frac{\pi}{\mathrm{2}} \\ $$