Question

Evaluate
$\left(i\right)\sin {60}^{\circ }\cos {30}^{\circ }+\sin {30}^{\circ }\cos {60}^{\circ }$
$\left(ii\right)\frac{5{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}$

• A. $\left(i\right)7\left(ii\right)\frac{5}{32}$
• B. $\left(i\right)2\left(ii\right)\frac{33}{8}$
• C. $\left(i\right)1\left(ii\right)\frac{67}{12}$
• D. $\left(i\right)5\left(ii\right)\frac{6}{47}$

Answer 1

The correct option is C $\left(i\right)1\left(ii\right)\frac{67}{12}$

$\left(i\right)\sin {60}^{\circ }\cos {30}^{\circ }+\sin {30}^{\circ }\cos {60}^{\circ }$
$=\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)={\left(\frac{\sqrt{3}}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2$
$=\frac{3}{4}+\frac{1}{4}=1$
$\left(ii\right)\frac{5{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}$
$=\frac{5(\cos {60}^{\circ }{)}^2+4(\sec {30}^{\circ }{)}^2-(\tan {45}^{\circ }{)}^2}{(\sin {30}^{\circ }{)}^2+(\cos {30}^{\circ }{)}^2}$
$=\frac{5{\left(\frac{1}{2}\right)}^2+4{\left(\frac{2}{\sqrt{3}}\right)}^2-(1{)}^2}{{\left(\frac{1}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}$
$=\frac{\frac{5}{4}+4\times \frac{4}{3}-1}{\frac{1}{4}+\frac{3}{4}}=\frac{\frac{5}{4}+\frac{16}{3}-1}{\frac{1}{4}+\frac{3}{4}}=\frac{5}{4}+\frac{16}{3}-1=\frac{15+64-12}{12}=\frac{67}{12}$