Question

The minimum value of $3{\tan }^2\theta +12{\cot }^2\theta$ is:

• A. $12$
• B. $15$
• C. $24$
• D. none of these

Answer 1

Answer: (A)

$12$

We have to find the minimum value of $3{\tan }^2\theta +12{\cot }^2\theta$, here both the terms are positive.
We apply the property A.M. > G.M. which says $\frac{a+b}{2}>\sqrt{ab}$
Thus, $\frac{3{\tan }^2\theta +12{\cot }^2\theta }{2}>\sqrt{3{\tan }^{2}theta\times 12{\cot }^2\theta }$
$\Rightarrow \frac{3{\tan }^2\theta +12{\cot }^2\theta }{2}>6$
$\Rightarrow 3{\tan }^2\theta +12{\cot }^2\theta >12$