The minimum value of `(sintheta+c o s e ctheta)^2+(costheta+sectheta)^2=`

Options

A. 7
B. 8
C. 9
D. none of these

Answer

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

### Step 1: Rewrite the Expression
We start by rewriting the expression using the definitions of cosecant and secant:
\[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can express the original equation as:
\[ (\sin \theta + \frac{1}{\sin \theta})^2 + (\cos \theta + \frac{1}{\cos \theta})^2 \]

### Step 2: Expand Each Square
Next, we expand both squares:
\[ (\sin \theta + \frac{1}{\sin \theta})^2 = \sin^2 \theta + 2 + \frac{1}{\sin^2 \theta} \]
\[ (\cos \theta + \frac{1}{\cos \theta})^2 = \cos^2 \theta + 2 + \frac{1}{\cos^2 \theta} \]
Combining these, we have:
\[ \sin^2 \theta + \cos^2 \theta + 4 + \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} \]

### Step 3: Use the Pythagorean Identity
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify the expression:
\[ 1 + 4 + \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} = 5 + \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} \]

### Step 4: Combine the Terms
We can combine the terms involving \(\sin^2 \theta\) and \(\cos^2 \theta\):
\[ \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\sin^2 \theta \cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]
Thus, we can rewrite our expression as:
\[ 5 + \frac{1}{\sin^2 \theta \cos^2 \theta} \]

### Step 5: Find the Minimum Value
To minimize the expression \(5 + \frac{1}{\sin^2 \theta \cos^2 \theta}\), we need to maximize \(\sin^2 \theta \cos^2 \theta\). The maximum value of \(\sin^2 \theta \cos^2 \theta\) occurs when \(\sin^2 \theta = \cos^2 \theta = \frac{1}{2}\):
\[ \sin^2 \theta \cos^2 \theta = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Thus, we have:
\[ \frac{1}{\sin^2 \theta \cos^2 \theta} = \frac{1}{\frac{1}{4}} = 4 \]

### Step 6: Final Calculation
Substituting this back into our expression gives:
\[ 5 + 4 = 9 \]

### Conclusion
Therefore, the minimum value of \((\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2\) is:
\[ \boxed{9} \]