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Question Number 145305 by puissant last updated on 04/Jul/21

$$\mathrm{R}\acute{\mathrm{e}soudre}\frac{\frac{8}{{\sin }^2\left(\mathrm{x}\right)}+1}{\frac{1}{{\cos }^2\left(\mathrm{x}\right)}+{\tan }^2\left(\mathrm{x}\right)}={\mathrm{cotan}}^2\left(\mathrm{x}\right)+\frac{4}{3}$$

Commented by imjagoll last updated on 04/Jul/21

$$\mathrm{put}\mathrm{u}=\cos {}^2\mathrm{x}$$

Commented by puissant last updated on 04/Jul/21

$$\mathrm{please}\mathrm{do}$$

Answered by imjagoll last updated on 04/Jul/21

$$\frac{8+\sin {}^2\mathrm{x}}{2\sec {}^2\mathrm{x}-1}=\cos {}^2\mathrm{x}+\frac{4\sin {}^2\mathrm{x}}{3}$$$$\Rightarrow \frac{9-\cos {}^2\mathrm{x}}{2\sec {}^2\mathrm{x}-1}=\frac{4-\cos {}^2\mathrm{x}}{3}$$$$\Rightarrow \frac{9-\mathrm{u}}{\frac{2}{\mathrm{u}}-1}=\frac{4-\mathrm{u}}{3}$$$$\Rightarrow \frac{9\mathrm{u}-{\mathrm{u}}^2}{2-\mathrm{u}}=\frac{4-\mathrm{u}}{3}$$$$\Rightarrow 27\mathrm{u}-{3\mathrm{u}}^2=8-6\mathrm{u}+{\mathrm{u}}^2$$$$\Rightarrow {4\mathrm{u}}^2-33\mathrm{u}+8=0$$$$\Rightarrow (4\mathrm{u}-1)(\mathrm{u}-2)=0$$$$\Rightarrow \mathrm{u}=\frac{1}{4}=\cos {}^2\mathrm{x}$$$$\Rightarrow \cos \mathrm{x}=\pm \frac{1}{2}$$

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