Minimum value of `sec ^(2) theta + cos ^(2) theta` is

Options

A. 1
B. 0
C. 2
D. none of these

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ f(\theta) = \sec^2 \theta + \cos^2 \theta \]
Recall that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). Thus, we can rewrite the function as:
\[ f(\theta) = \frac{1}{\cos^2 \theta} + \cos^2 \theta \]

### Step 2: Differentiate the function
To find the minimum value, we need to differentiate \( f(\theta) \) with respect to \( \theta \):
\[ f'(\theta) = -\frac{2 \sin \theta}{\cos^3 \theta} + 2 \cos \theta \]
Setting the derivative equal to zero for critical points:
\[ -\frac{2 \sin \theta}{\cos^3 \theta} + 2 \cos \theta = 0 \]

### Step 3: Solve for critical points
Rearranging gives:
\[ 2 \cos \theta = \frac{2 \sin \theta}{\cos^3 \theta} \]
Multiplying both sides by \( \cos^3 \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ 2 \cos^4 \theta = 2 \sin \theta \]
This simplifies to:
\[ \cos^4 \theta = \sin \theta \]

### Step 4: Use Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin \theta \) as:
\[ \sin \theta = 1 - \cos^2 \theta \]
Substituting this into our equation gives:
\[ \cos^4 \theta = 1 - \cos^2 \theta \]
Let \( x = \cos^2 \theta \). Then we have:
\[ x^2 + x - 1 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]
Since \( x = \cos^2 \theta \) must be non-negative, we take:
\[ x = \frac{-1 + \sqrt{5}}{2} \]

### Step 6: Calculate \( f(\theta) \)
Now substituting \( \cos^2 \theta \) back into \( f(\theta) \):
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} = \frac{2}{-1 + \sqrt{5}} \]
Thus:
\[ f(\theta) = \frac{2}{-1 + \sqrt{5}} + \frac{-1 + \sqrt{5}}{2} \]
Calculating this gives:
\[ f(\theta) = 2 \]

### Conclusion
The minimum value of \( \sec^2 \theta + \cos^2 \theta \) is:
\[ \boxed{2} \]