Question Number 117497 by bemath last updated on 12/Oct/20
$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{equation}\:\mathrm{sec}\:\mathrm{3}\theta\:=\:\mathrm{sec}\:\theta \\ $$
Answered by Dwaipayan Shikari last updated on 12/Oct/20
$$\frac{\mathrm{1}}{{cos}\mathrm{3}\theta}=\frac{\mathrm{1}}{{cos}\theta} \\ $$$${cos}\mathrm{3}\theta={cos}\theta \\ $$$$\mathrm{4}{cos}^{\mathrm{3}} \theta−\mathrm{3}{cos}\theta={cos}\theta \\ $$$${cos}^{\mathrm{3}} \theta={cos}\theta \\ $$$${cos}\theta\left({cos}^{\mathrm{2}} \theta−\mathrm{1}\right)=\mathrm{0} \\ $$$${cos}\theta=\mathrm{0} \\ $$$${cos}^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{0} \\ $$$$\theta={k}\pi+\frac{\pi}{\mathrm{2}} \\ $$$${cos}\theta=\mathrm{1}\:\:\:{or}\:\:{cos}\theta=−\mathrm{1} \\ $$$$\theta=\mathrm{2}{k}\pi\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\theta=\mathrm{2}\pi{k}+\pi\:\:\:\:\:\left({k}\in\mathbb{Z}\right) \\ $$