Question

The minimum value of 9 ${\tan }^2\theta +4{\cot }^2\theta$is

• A. 13
• B. 9
• C. 6
• D. 12

Answer 1

The correct option is D 12

$f\left(x\right)=9{\tan }^2\theta +4{\cot }^2\theta$
put $y={\tan }^2\theta$
$f\left(y\right)=9y+\frac{4}{y}$
diff $f\left(y\right)$ with respect to y, for maxima or minima evolution.
$f^{\prime }\left(y\right)=9-\frac{4}{y^2}=0$
$y=\frac{2}{3},-\frac{2}{3}(-\frac{2}{3}rejected)$
$\therefore$ minimum value of $f\left(x\right)=9\times \frac{2}{3}+4\times \frac{3}{2}=12$