Find minimum value of `4cos^(2)theta+9sec^(2)theta`

Answer

To find the minimum value of the expression \(4\cos^2\theta + 9\sec^2\theta\), we will use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ y = 4\cos^2\theta + 9\sec^2\theta \]
Recall that \(\sec^2\theta = \frac{1}{\cos^2\theta}\). Therefore, we can rewrite the expression as:
\[ y = 4\cos^2\theta + 9\frac{1}{\cos^2\theta} \]

2. **Apply AM-GM Inequality**:
By the AM-GM inequality, for any non-negative \(a\) and \(b\):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
We can apply this to our expression. Let:
\[ a = 4\cos^2\theta \quad \text{and} \quad b = 9\sec^2\theta \]
Then:
\[ \frac{4\cos^2\theta + 9\sec^2\theta}{2} \geq \sqrt{4\cos^2\theta \cdot 9\sec^2\theta} \]

3. **Calculate the Geometric Mean**:
Calculate the right-hand side:
\[ \sqrt{4\cos^2\theta \cdot 9\sec^2\theta} = \sqrt{36} = 6 \]
Thus, we have:
\[ \frac{4\cos^2\theta + 9\sec^2\theta}{2} \geq 6 \]

4. **Multiply by 2**:
Multiplying both sides by 2 gives:
\[ 4\cos^2\theta + 9\sec^2\theta \geq 12 \]

5. **Conclusion**:
The minimum value of \(4\cos^2\theta + 9\sec^2\theta\) is therefore:
\[ \text{Minimum value} = 12 \]