Question

Evaluate $\frac{1}{4}\left(co{t}^{4}30°-\cos e{c}^{4}60°\right)+\frac{3}{2}\left(se{c}^{2}45°-{\tan }^{2}30°\right)-5{\cos }^{2}60°$.

• A. $\frac{55}{18}$
• B. $\frac{11}{6}$
• C. $\frac{7}{4}$
• D. $0$

Answer 1

The correct option is A

$\frac{55}{18}$

Explanation of the correct option:

Find the value of expression,

$$\begin{gathered} \frac{1}{4}\left(co{t}^{4}30°-\cos e{c}^{4}60°\right)+\frac{3}{2}\left(se{c}^{2}45°-{\tan }^{2}30°\right)-5{\cos }^{2}60° \\ =\frac{1}{4}\left({\left(\cot 30°\right)}^4-{\left(\mathrm{cosec}60°\right)}^4\right)+\frac{3}{2}\left({\left(\sec 45°\right)}^2-{\left(\tan 30°\right)}^2\right)-5{\left(\cos 60°\right)}^2 \end{gathered}$$

We know the values of $\cot 30°=\sqrt{3},\mathrm{cosec}60°=\frac{2}{\sqrt{3}},\sec 45°=\sqrt{2},\tan 30°=\frac{1}{\sqrt{3}}$ and $\cos 60°=\frac{1}{2},$.

So, by substituting these values,

$$\begin{gathered} =\frac{1}{4}\left[{\left(\sqrt{3}\right)}^4-{\left(\frac{2}{\sqrt{3}}\right)}^4\right]+\frac{3}{2}\left[{\left(\sqrt{2}\right)}^2-{\left(\frac{1}{\sqrt{3}}\right)}^2\right]-5{\left(\frac{1}{2}\right)}^2 \\ =\frac{1}{4}\left(9-\frac{16}{9}\right)+\frac{3}{2}\left(2-\frac{1}{3}\right)-5\times \frac{1}{4} \\ =\frac{1}{4}\left(\frac{81-16}{9}\right)+\frac{3}{2}\left(\frac{6-1}{3}\right)-\frac{5}{4} \\ =\frac{1}{4}\times \frac{65}{9}+\frac{3}{2}\times \frac{5}{3}-\frac{5}{4} \\ =\frac{65}{36}+\frac{5}{2}-\frac{5}{4} \\ =\frac{65+90-45}{36} \\ =\frac{110}{36} \\ =\frac{55}{18} \end{gathered}$$

Hence, the correct option is $\mathrm{A}$.