Question Number 113846 by deepraj123 last updated on 15/Sep/20
$$\mathrm{Thr}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{tan}\:\mathrm{3}{x}=\mathrm{tan}\:\mathrm{5}{x}\:\:\mathrm{is} \\ $$
Answered by MJS_new last updated on 15/Sep/20
$${x}=\mathrm{0}+{n}\pi={n}\pi;\:{n}\in\mathbb{Z} \\ $$
Answered by 1549442205PVT last updated on 16/Sep/20
$$\mathrm{tan3x}=\mathrm{tan5x}\Leftrightarrow\mathrm{3x}=\mathrm{5x}+\mathrm{k}\pi \\ $$$$\Leftrightarrow\mathrm{x}=\mathrm{k}\frac{\pi}{\mathrm{2}}\left(\mathrm{k}\in\mathbb{Z}\right) \\ $$
Commented by MJS_new last updated on 16/Sep/20
$$\mathrm{both}\:\mathrm{tan}\:\frac{\mathrm{3}{k}\pi}{\mathrm{2}}\:\mathrm{and}\:\mathrm{tan}\:\frac{\mathrm{5k}\pi}{\mathrm{2}}\:\mathrm{are}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{if} \\ $$$${k}=\mathrm{2}{n}+\mathrm{1}\:\Rightarrow\:{k}=\mathrm{2}{n}\:\Rightarrow\:{x}={n}\pi \\ $$