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Question Number 117497 by bemath last updated on 12/Oct/20

find the solution set of the   equation sec 3θ = sec θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{equation}\:\mathrm{sec}\:\mathrm{3}\theta\:=\:\mathrm{sec}\:\theta \\ $$

Answered by Dwaipayan Shikari last updated on 12/Oct/20

(1/(cos3θ))=(1/(cosθ))  cos3θ=cosθ  4cos^3 θ−3cosθ=cosθ  cos^3 θ=cosθ  cosθ(cos^2 θ−1)=0  cosθ=0  cos^2 θ−1=0  θ=kπ+(π/2)  cosθ=1   or  cosθ=−1  θ=2kπ                 θ=2πk+π     (k∈Z)

$$\frac{\mathrm{1}}{{cos}\mathrm{3}\theta}=\frac{\mathrm{1}}{{cos}\theta} \\ $$$${cos}\mathrm{3}\theta={cos}\theta \\ $$$$\mathrm{4}{cos}^{\mathrm{3}} \theta−\mathrm{3}{cos}\theta={cos}\theta \\ $$$${cos}^{\mathrm{3}} \theta={cos}\theta \\ $$$${cos}\theta\left({cos}^{\mathrm{2}} \theta−\mathrm{1}\right)=\mathrm{0} \\ $$$${cos}\theta=\mathrm{0} \\ $$$${cos}^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{0} \\ $$$$\theta={k}\pi+\frac{\pi}{\mathrm{2}} \\ $$$${cos}\theta=\mathrm{1}\:\:\:{or}\:\:{cos}\theta=−\mathrm{1} \\ $$$$\theta=\mathrm{2}{k}\pi\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\theta=\mathrm{2}\pi{k}+\pi\:\:\:\:\:\left({k}\in\mathbb{Z}\right) \\ $$

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