If `f(theta)=(sintheta+cos e ctheta)^2+(costheta+s e ctheta)^2,` then minimum value of `f(theta)` is `7` `8` `9` 4. None of these

Explanation

To find the minimum value of the function \( f(\theta) = (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 \), we can follow these steps:

### Step 1: Expand the squares
We start by expanding the squares in the function:

\[ f(\theta) = (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 \]

Expanding each term:

\[ = \sin^2 \theta + 2\sin \theta \csc \theta + \csc^2 \theta + \cos^2 \theta + 2\cos \theta \sec \theta + \sec^2 \theta \]

### Step 2: Simplify the terms
We know that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Therefore:

\[ 2\sin \theta \csc \theta = 2 \quad \text{and} \quad 2\cos \theta \sec \theta = 2 \]

Now substituting these back into the equation:

\[ f(\theta) = \sin^2 \theta + \cos^2 \theta + 2 + \csc^2 \theta + \sec^2 \theta + 2 \]

### Step 3: Use Pythagorean identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ f(\theta) = 1 + 4 + \csc^2 \theta + \sec^2 \theta \]

### Step 4: Express \( \csc^2 \theta \) and \( \sec^2 \theta \)
We know that:

\[ \csc^2 \theta = 1 + \cot^2 \theta \quad \text{and} \quad \sec^2 \theta = 1 + \tan^2 \theta \]

Thus:

\[ f(\theta) = 5 + \cot^2 \theta + \tan^2 \theta \]

### Step 5: Use AM-GM inequality
Using the AM-GM inequality on \( \cot^2 \theta \) and \( \tan^2 \theta \):

\[ \cot^2 \theta + \tan^2 \theta \geq 2 \]

This gives us:

\[ f(\theta) \geq 5 + 2 = 7 \]

### Step 6: Find the minimum value
To find the minimum value of \( f(\theta) \), we note that the minimum occurs when \( \cot^2 \theta = \tan^2 \theta = 1 \), which happens when \( \theta = \frac{\pi}{4} \) (or any angle where sine and cosine are equal).

Substituting back, we find:

\[ f(\theta) = 5 + 2 = 7 \]

### Conclusion
Thus, the minimum value of \( f(\theta) \) is:

\[ \boxed{9} \]