Question

The value of $({\sin }^{2}30°{\cos }^{2}45°+4{\tan }^{2}30°+\frac{1}{2}{\sin }^{2}90°+\frac{1}{8}{\cot }^{2}60°)=?$
(a) $\frac{3}{8}$
(b) $\frac{5}{8}$
(c) 6
(d) 2

Answer 1

(d) 2
$$\begin{gathered} \left({\sin }^2{30}^0{\cos }^2{45}^0\right)+4{\tan }^2{30}^0+\frac{1}{2}{\sin }^2{90}^0+\frac{1}{8}{\cot }^2{60}^0 \\ \text{=}\frac{1}{2^2}\times \frac{1}{{\left(\sqrt{2}\right)}^2}+4\times \frac{1}{{\left(\sqrt{3}\right)}^2}+\frac{1}{2}\times 1^2+\frac{1}{8}\times \frac{1}{{\left(\sqrt{3}\right)}^2}\left[\begin{array}{l}\because \sin {30}^0=\frac{1}{2}\text{and}\cos {45}^0=\frac{1}{\sqrt{2}}\\ {\text{and tan30}}^0=\frac{1}{2}\text{and}\cot {60}^0=\frac{1}{\sqrt{3}}\end{array}\right] \\ \text{=}\frac{1}{4}\times \frac{1}{2}+4\times \frac{1}{3}+\frac{1}{2}+\frac{1}{24} \\ \text{=}\frac{1}{8}+\frac{4}{3}+\frac{1}{2}+\frac{1}{24} \\ \text{=}\frac{3+32+12+1}{24} \\ \text{=}\frac{48}{24} \\ \text{=}2 \end{gathered}$$