Find the minimum value of `sec^2 theta + cosec^2 theta - 4.`

Answer

To find the minimum value of the expression \( \sec^2 \theta + \csc^2 \theta - 4 \), we can follow these steps:

### Step 1: Define the function
Let \( f(\theta) = \sec^2 \theta + \csc^2 \theta - 4 \).

### Step 2: Differentiate the function
To find the critical points, we need to differentiate \( f(\theta) \):
\[ f'(\theta) = \frac{d}{d\theta}(\sec^2 \theta) + \frac{d}{d\theta}(\csc^2 \theta) \]
Using the derivatives:
- The derivative of \( \sec^2 \theta \) is \( 2 \sec^2 \theta \tan \theta \).
- The derivative of \( \csc^2 \theta \) is \( -2 \csc^2 \theta \cot \theta \).

Thus, we have:
\[ f'(\theta) = 2 \sec^2 \theta \tan \theta - 2 \csc^2 \theta \cot \theta \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 2 \sec^2 \theta \tan \theta - 2 \csc^2 \theta \cot \theta = 0 \]
Dividing through by 2 gives:
\[ \sec^2 \theta \tan \theta = \csc^2 \theta \cot \theta \]

### Step 4: Use trigonometric identities
Recall that:
- \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \)
- \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Substituting these into the equation gives:
\[ \frac{1}{\cos^2 \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\sin^2 \theta} \cdot \frac{\cos \theta}{\sin \theta} \]
This simplifies to:
\[ \frac{\sin \theta}{\cos^3 \theta} = \frac{\cos \theta}{\sin^3 \theta} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sin^4 \theta = \cos^4 \theta \]
This implies:
\[ \tan^4 \theta = 1 \]
Taking the fourth root gives:
\[ \tan \theta = 1 \quad \text{or} \quad \tan \theta = -1 \]
Thus, \( \theta = \frac{\pi}{4} + n\pi \) for integers \( n \).

### Step 6: Evaluate the function at critical points
We will evaluate \( f(\theta) \) at \( \theta = \frac{\pi}{4} \):
\[ \sec^2\left(\frac{\pi}{4}\right) = 2, \quad \csc^2\left(\frac{\pi}{4}\right) = 2 \]
Thus:
\[ f\left(\frac{\pi}{4}\right) = 2 + 2 - 4 = 0 \]

### Conclusion
The minimum value of \( \sec^2 \theta + \csc^2 \theta - 4 \) is \( 0 \).