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  4. The minimum value of 27 tan 2 θ+3 cot 2 θ is

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Q. The minimum value of $27 \tan ^{2} \theta+3 \cot ^{2} \theta$ is

EAMCETEAMCET 2012

Solution:

Given trigonometrical equation is
$27 \tan ^{2} \theta+3 \cot ^{2} \theta$ (Using $AM \ge GM$)
$ \therefore \frac{27 \tan ^{2} \theta+3 \cot ^{2} \theta}{2} \geq \sqrt{27 \tan ^{2} \theta \cdot 3 \cot ^{2} \theta}$
$\Rightarrow \frac{27 \tan ^{2} \theta+3 \cot ^{2} \theta}{2} \geq \sqrt{81}$
$\Rightarrow 27 \tan ^{2} \theta+3 \cot ^{2} \theta \geq 18$
Hence, minimum value of given equation is 18