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

$12$

Calculate the value of the expression.

The given expression is $9{\tan }^2\theta +4{\cot }^2\theta$.

$$\begin{gathered} ={\left(3\tan \theta \right)}^2+{\left(2cot\theta \right)}^2 \\ ={\left(3\tan \theta -2\cot \theta \right)}^2+2·3\tan \theta ·2\cot \theta \left[\because a^2+b^2={\left(a-b\right)}^2+2ab\right] \\ ={\left(3\tan \theta -2\cot \theta \right)}^2+12\left[\because tan\left(\theta \right)cot\left(\theta \right)=1\right] \end{gathered}$$

Since $3\tan \theta -2\cot \theta$ is a square term then for any $\theta \in \mathrm{ℝ}$,${\left(3\tan \theta -2\cot \theta \right)}^2\ge 0$

Clearly for minimum $3\tan \theta -2\cot \theta =0$

Therefore the minimum value of $9{\tan }^2\theta +4{\cot }^2\theta$ is $12$

Hence, option D is correct.