Question

Which one of the following is not rational -`

• A. $\frac{1-{\tan }^2{30}^{\circ }}{1+{\tan }^2{30}^{\circ }}$
• B. $4{\cos }^3{30}^{\circ }-3\cos {30}^{\circ }$
• C. $3\sin {30}^{\circ }-4{\sin }^3{30}^{\circ }$
• D. $\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}$

Answer 1

Answer: (D)

$\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}$

(A) $\frac{1-{\tan }^2{30}^{\circ }}{1+{\tan }^2{30}^{\circ }}=\frac{1-\frac{1}{3}}{1+\frac{1}{3}}=\frac{\frac{2}{3}}{\frac{4}{3}}=\frac{1}{2}$
(B) $4{\cos }^2{30}^{\circ }-2\cos {30}^{\circ }=4{\left(\frac{\sqrt{3}}{2}\right)}^3-3\left(\frac{\sqrt{3}}{2}\right)$
$=\frac{4\times 3}{4}\frac{\sqrt{3}}{2}-\frac{3\sqrt{3}}{2}=0$
(C) $3\sin {30}^{\circ }-4{\sin }^3{30}^{\circ }=\frac{3}{2}-4{\left(\frac{1}{2}\right)}^3=\frac{3}{2}-\frac{4}{8}=1$
(D) $\frac{2\cot {30}^{\circ }}{{\cot }^2{30}^{\circ }-1}=\frac{2\sqrt{3}}{3-1}=\sqrt{3}$ (not rational)