Question

$\left({\sin }^{2}30°{\cos }^{2}45°+4{\tan }^{2}30°+\frac{1}{2}{\sin }^{2}90°+\frac{1}{8}{\cot }^{2}60°\right)=?$
(a) 2
(b) 3
(c) 4
(d) $\frac{5}{2}$

Answer 1

(a) 2


$$\begin{gathered} \left({\sin }^2{30}^{\mathrm{o}}{\cos }^2{45}^{\mathrm{o}}+4{\tan }^2{30}^{\mathrm{o}}+\frac{1}{2}{\sin }^2{90}^{\mathrm{o}}+\frac{1}{8}{\cot }^2{60}^{\mathrm{o}}\right) \\ =\left[{\left(\frac{1}{2}\right)}^2\times {\left(\frac{1}{\sqrt{2}}\right)}^2+4\times {\left(\frac{1}{\sqrt{3}}\right)}^2+\frac{1}{2}\times {\left(1\right)}^2+\frac{1}{8}\times {\left(\frac{1}{\sqrt{3}}\right)}^2\right] \\ =\left[\left({\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{2}}\right)+\frac{4}{3}+\frac{1}{2}+\frac{1}{24}\right] \\ =\left[\frac{1}{8}+\frac{4}{3}+\frac{1}{2}+\frac{1}{24}\right] \\ =\frac{3+32+12+1}{24} \\ =\frac{48}{24}=2 \end{gathered}$$