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Question Number 164543 by cortano1 last updated on 18/Jan/22

((4(√(1−x)))/x) +((√(2x−1))/(1−x)) = 3(√3)

$$\:\:\frac{\mathrm{4}\sqrt{\mathrm{1}−{x}}}{{x}}\:+\frac{\sqrt{\mathrm{2}{x}−\mathrm{1}}}{\mathrm{1}−{x}}\:=\:\mathrm{3}\sqrt{\mathrm{3}} \\ $$

Answered by leonhard77 last updated on 19/Jan/22

(1) 1−x>0 ∩ 2x−1≥0 (1/2)≤x<1 (2) let (√x) =cos t ((4sin t)/(cos ^2 t)) + ((√(cos 2t))/(sin^2 t)) = 3(√3) 4sin^3 t + cos^2 t (√(cos 2t)) = ((3(√3))/4) sin^2 2t

$$\:\left(\mathrm{1}\right)\:\mathrm{1}−{x}>\mathrm{0}\:\cap\:\mathrm{2}{x}−\mathrm{1}\geqslant\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\leqslant{x}<\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{let}\:\sqrt{{x}}\:=\mathrm{cos}\:\:{t} \\ $$$$\:\:\frac{\mathrm{4sin}\:\:{t}}{\mathrm{cos}\:\:^{\mathrm{2}} {t}}\:+\:\frac{\sqrt{\mathrm{cos}\:\mathrm{2}{t}}}{\mathrm{sin}\:^{\mathrm{2}} {t}}\:=\:\mathrm{3}\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{4sin}\:^{\mathrm{3}} {t}\:+\:\mathrm{cos}\:^{\mathrm{2}} {t}\:\sqrt{\mathrm{cos}\:\mathrm{2}{t}}\:=\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{4}}\:\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{t} \\ $$$$\: \\ $$

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