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The minimum value of $27 \tan ^2 \theta+3 \cot ^2 \theta$ is
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The correct answer is:
$18$
Given trigonometrical equation is
$27 \tan ^2 \theta+3 \cot ^2 \theta$
$\begin{aligned} & \therefore \quad \frac{27 \tan ^2 \theta+3 \cot ^2 \theta}{2} \geq \sqrt{27 \tan ^2 \theta \cdot 3 \cot ^2 \theta} \\ & \Rightarrow \frac{27 \tan ^2 \theta+3 \cot ^2 \theta}{2} \geq \sqrt{81} \\ & \Rightarrow 27 \tan ^2 \theta+3 \cot ^2 \theta \geq 18\end{aligned}$
Hence, minimum value of given equation is 18
$27 \tan ^2 \theta+3 \cot ^2 \theta$
$\begin{aligned} & \therefore \quad \frac{27 \tan ^2 \theta+3 \cot ^2 \theta}{2} \geq \sqrt{27 \tan ^2 \theta \cdot 3 \cot ^2 \theta} \\ & \Rightarrow \frac{27 \tan ^2 \theta+3 \cot ^2 \theta}{2} \geq \sqrt{81} \\ & \Rightarrow 27 \tan ^2 \theta+3 \cot ^2 \theta \geq 18\end{aligned}$
Hence, minimum value of given equation is 18
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