Question Number 14268 by Tinkutara last updated on 30/May/17
$$\mathrm{If}\:\mathrm{tan}\theta\:+\:\mathrm{tan2}\theta\:=\:\mathrm{tan3}\theta,\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{exhaustive}\:\mathrm{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\theta\:\mathrm{satisfying} \\ $$$$\mathrm{the}\:\mathrm{given}\:\mathrm{equation}. \\ $$
Answered by linkelly0615 last updated on 30/May/17
$$\because{tan}\mathrm{3}\theta=\frac{{tan}\theta+{tan}\mathrm{2}\theta}{\mathrm{1}−{tan}\theta{tan}\mathrm{2}\theta}={tan}\theta+{tan}\mathrm{2}\theta \\ $$$$\therefore\mathrm{1}−{tan}\theta{tan}\mathrm{2}\theta=\mathrm{1} \\ $$$$\therefore{tan}\theta{tan}\mathrm{2}\theta=\mathrm{0}\:,{and}\:\theta\notin\left({n}+\mathrm{1}/\mathrm{2}\right)\pi\:,\:{n}\in\mathbb{Z} \\ $$$$\therefore\theta={k}\pi\:,\:{k}\in\mathbb{Z}# \\ $$
Commented by Tinkutara last updated on 30/May/17
$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{complete}\:\mathrm{set}\:\mathrm{of} \\ $$$$\mathrm{solution}? \\ $$
Commented by prakash jain last updated on 30/May/17
$${another}\:{possibility}\:{is} \\ $$$$\mathrm{tan}\:\theta+\mathrm{tan}\:\mathrm{2}\theta=\mathrm{0} \\ $$