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Question: The minimum value of \(9\tan^{2}\theta + 4\cot^{2}\theta\)is...

The minimum value of 9tan⁡2θ+4cot⁡2θ9\tan^{2}\theta + 4\cot^{2}\thetais

A

13

B

9

C

6

D

12

Answer

12

Explanation

Solution

A.M. ≥\geq G.M.⇒9tan⁡2θ+4cot⁡2θ2≥4cot⁡2θ.9tan⁡2θ\Rightarrow \frac{9\tan^{2}\theta + 4\cot^{2}\theta}{2} \geq \sqrt{4\cot^{2}\theta.9\tan^{2}\theta}

⇒9tan⁡2θ+4cot⁡2θ≥12\Rightarrow 9\tan^{2}\theta + 4\cot^{2}\theta \geq 12

Therefore, the minimum value is 12.