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Question Number 75901 by Master last updated on 20/Dec/19

Answered by MJS last updated on 20/Dec/19

$$\frac{1}{{\cos }^{2}x}=4\tan 2x$$$$\frac{1}{1+\cos 2x}=2\tan 2x$$$$x=\arctan t$$$$\frac{t^2+1}{2}=\frac{4t}{1-t^2}$$$$t^4+8t-1=0$$$$t_1\approx -2.04004$$$$t_2\approx .124970$$$$t_{3,4}\approx .957536\pm 1.73366\mathrm{i}$$$$\Rightarrow$$$$x=n\pi -1.11503\vee x=n\pi +.124325\mathrm{with}n\in \mathbb{Z}$$

Commented by Master last updated on 21/Dec/19

Commented by MJS last updated on 21/Dec/19

$$\mathrm{believe}\mathrm{me}\mathrm{or}\mathrm{not}$$$$\mathrm{try}\mathrm{the}\mathrm{given}\mathrm{options},\mathrm{they}\mathrm{are}\mathrm{all}\mathrm{wrong}$$

Commented by MJS last updated on 21/Dec/19

$$\mathrm{I}\mathrm{guess}\mathrm{the}\mathrm{question}\mathrm{is}\mathrm{wrong}$$$$\frac{1}{{\cos }^{2}x}={4\tan }^{2}x$$$$\frac{1}{{\cos }^{2}x}=4\frac{{\sin }^{2}x}{{\cos }^{2}x}$$$${\cos }^{2}x\ne 0\Rightarrow x\ne \frac{\pi }{2}+n\pi$$$${\sin }^{2}x=\frac{1}{4}$$$$\sin x=\pm \frac{1}{2}$$$$x=\pm \frac{\pi }{6}+n\pi$$

Commented by Master last updated on 21/Dec/19

$$\mathrm{thanks}$$

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