If cos \(\theta\) = \(\frac{\sqrt{3}}{2}\) and \(\theta\) is less than 90o. Calculate \(\frac{\cot(90 - \theta)}{sin^2\theta}\)
A
\(\frac{4}{\sqrt{3}}\)
B
\(4 \sqrt{3}\)
C
\(\sqrt{\frac{3}{2}}\)
D
\(\frac{1}{\sqrt{3}}\)
E
\(\frac{2}{\sqrt{3}}\)
Verified Correct Answer: A.
Step-by-Step Mathematical Solution
\(\frac{\cot (90 - \theta)}{sin^2\theta}\)
\(\cot (90 - \theta) = \tan \theta\)
\(\therefore \frac{\cot (90 - \theta)}{\sin^{2} \theta} = \frac{\tan \theta}{\sin^{2} \theta}\)
\(\tan \theta = \frac{\sqrt{3}}{3}\)
\(\sin \theta = \frac{1}{2} \implies \sin^{2} \theta = \frac{1}{4}\)
\(\frac{\cot(90 - \theta)}{\sin^{2} \theta} = \frac{\sqrt{3}}{3}\div\frac{1}{4}\)
= \(\frac{4}{\sqrt{3}}\)
Source: Myschool Past Questions Archive (Record #32268)