Cho góc \(x\left(0^{\circ} \leq x \leq 180^{\circ}\right)\) thỏa mãn \(\cos x=\frac{1}{4}\). Giá trị của \(P=\frac{\tan ^{2} x-\tan x+3 \cot x}{\frac{1}{2 \cos x}-5 \tan x+30 \cot ^{2} x}\) là
\(\frac{1}{\sqrt{5}}\)
\(\sqrt{\frac{3}{5}}\).
\(\frac{1}{\sqrt{15}}\).
\(-\sqrt{\frac{3}{5}}\).
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Ta có: \(\cos x=\frac{1}{4}\)
\(\Rightarrow \sin x=\sqrt{1-\cos ^{2} x}=\sqrt{1-\left(\frac{1}{4}\right)^{2}}=\frac{\sqrt{15}}{4} ;\)
\(\tan x=\frac{\sin x}{\cos x}=\frac{\frac{\sqrt{15}}{4}}{\frac{1}{4}}=\sqrt{15} ;\)
\(\cot x=\frac{1}{\tan x}=\frac{1}{\sqrt{15}}\).
\(\Rightarrow P=\frac{15-\sqrt{15}+\frac{3}{\sqrt{15}}}{\frac{1}{2 \cdot \frac{1}{4}}-5 \sqrt{15}+30 \cdot \frac{1}{15}}=-\sqrt{\frac{3}{5}}\).
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