The minimum value of $27~tan^{2} heta+3~cot^{2} heta$ is:

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

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

Question

The minimum value of $27~tan^{2}	heta+3~cot^{2}	heta$ is:

Accepted Answer

To determine the minimum value of the expression $27 \tan^2 \theta + 3 \cot^2 \theta$, we need to analyze the behavior of the trigonometric functions involved.

First, recall the identity between tangent and cotangent:
$$
\tan \theta = \frac{1}{\cot \theta}
$$
Thus, we can express $\tan^2 \theta$ in terms of $\cot^2 \theta$:
$$
\tan^2 \theta = \frac{1}{\cot^2 \theta}
$$

Substituting this into the original expression, we have:
$$
27 \tan^2 \theta + 3 \cot^2 \theta = 27 \left(\frac{1}{\cot^2 \theta}\right) + 3 \cot^2 \theta
$$

Let $x = \cot^2 \theta$. Then, the expression becomes:
$$
27 \left(\frac{1}{x}\right) + 3x = \frac{27}{x} + 3x
$$

To find the minimum value of this expression, we can use calculus or the AM-GM inequality. Here, we'll use the AM-GM inequality, which states that for non-negative numbers $a$ and $b$, the arithmetic mean is greater than or equal to the geometric mean:
$$
\frac{a + b}{2} \geq \sqrt{ab}
$$

Applying this to our expression:
$$
\frac{\frac{27}{x} + 3x}{2} \geq \sqrt{\frac{27}{x} \cdot 3x}
$$

Simplifying the right side:
$$
\sqrt{\frac{27}{x} \cdot 3x} = \sqrt{81} = 9
$$

Thus:
$$
\frac{\frac{27}{x} + 3x}{2} \geq 9
$$

Multiplying both sides by 2 gives:
$$
\frac{27}{x} + 3x \geq 18
$$

The equality holds when $\frac{27}{x} = 3x$, which simplifies to:
$$
27 = 3x^2 \quad \Rightarrow \quad x^2 = 9 \quad \Rightarrow \quad x = 3
$$

Substituting $x = 3$ back into the expression:
$$
\frac{27}{3} + 3 \times 3 = 9 + 9 = 18
$$

Therefore, the minimum value of the expression is indeed 18, confirming that Option 2 is correct.

**Explanation of Incorrect Options:**

1. **Option 1 (9):** This value is incorrect because the minimum value derived from the AM-GM inequality and verified by substitution is 18, not 9.

3. **Option 3 (27):** This value is incorrect because the calculated minimum value of the expression is 18, which is less than 27.

4. **Option 4 (30):** This value is incorrect because the expression achieves a minimum value of 18, which is less than 30.

Thus, the correct answer is Option 2, with the minimum value of the expression being 18.