Find min value of `17sec^(2)theta+15tan^(2)theta`.

Answer

To find the minimum value of the expression \( 17 \sec^2 \theta + 15 \tan^2 \theta \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Let \( x = \tan^2 \theta \). Then, we can express \( \sec^2 \theta \) in terms of \( x \):
\[ \sec^2 \theta = 1 + x \]

### Step 2: Substitute into the Expression
Now substitute \( \sec^2 \theta \) into the original expression:
\[ 17 \sec^2 \theta + 15 \tan^2 \theta = 17(1 + x) + 15x \]
This simplifies to:
\[ 17 + 17x + 15x = 17 + 32x \]

### Step 3: Analyze the Expression
The expression \( 17 + 32x \) is linear in terms of \( x \). Since \( x = \tan^2 \theta \) and \( \tan^2 \theta \geq 0 \), the minimum value occurs when \( x = 0 \).

### Step 4: Find the Minimum Value
Substituting \( x = 0 \) into the expression:
\[ 17 + 32(0) = 17 \]

### Conclusion
Thus, the minimum value of \( 17 \sec^2 \theta + 15 \tan^2 \theta \) is:
\[ \boxed{17} \]