Find minimum value of `25tan^(2)theta+4cot^(2)theta`.

Answer

To find the minimum value of the expression \( 25 \tan^2 \theta + 4 \cot^2 \theta \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ 25 \tan^2 \theta + 4 \cot^2 \theta \]

2. **Apply AM-GM Inequality**:
According to the AM-GM inequality:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
where \( a = 25 \tan^2 \theta \) and \( b = 4 \cot^2 \theta \).

3. **Set Up the Inequality**:
Applying the AM-GM inequality:
\[ \frac{25 \tan^2 \theta + 4 \cot^2 \theta}{2} \geq \sqrt{25 \tan^2 \theta \cdot 4 \cot^2 \theta} \]

4. **Simplify the Right Side**:
Simplifying the right-hand side:
\[ \sqrt{25 \tan^2 \theta \cdot 4 \cot^2 \theta} = \sqrt{100 \tan^2 \theta \cot^2 \theta} = \sqrt{100} = 10 \]

5. **Multiply by 2**:
Thus, we have:
\[ 25 \tan^2 \theta + 4 \cot^2 \theta \geq 2 \cdot 10 = 20 \]

6. **Conclusion**:
The minimum value of \( 25 \tan^2 \theta + 4 \cot^2 \theta \) is therefore:
\[ \boxed{20} \]