Question

$\left(\frac{4}{3}{\cot }^{2}30°+3{\sin }^{2}60°-2\cos e{c}^{2}60°-\frac{3}{4}{\tan }^{2}30°\right)=?$
(a) 3
(b) $\frac{8}{3}$
(c) $\frac{10}{3}$
(d) $\frac{9}{4}$

Answer 1

(c) $\frac{10}{3}$

$$\begin{gathered} \left(\frac{4}{3}{\cot }^2{30}^{\mathrm{o}}+3{\sin }^2{60}^{\mathrm{o}}-2{\mathrm{cosec}}^2{60}^{\mathrm{o}}-\frac{3}{4}{\tan }^2{30}^{\mathrm{o}}\right) \\ =\left[\frac{4}{3}\times {\left(\sqrt{3}\right)}^2+3\times {\left(\frac{\sqrt{3}}{2}\right)}^2-2\times {\left(\frac{2}{\sqrt{3}}\right)}^2-\frac{3}{4}\times {\left(\frac{1}{\sqrt{3}}\right)}^2\right] \\ =\left[4+\frac{9}{4}-\frac{8}{3}-\frac{1}{4}\right] \\ =\frac{48+27-32-3}{12} \\ =\frac{75-35}{12} \\ =\frac{40}{12} \\ =\frac{10}{3} \end{gathered}$$