What is the value of `{(1-sin^2theta) sec^2theta + tan^2theta} (cos^2theta + 1)`, where `0^@ lt theta lt 90^@`?

Options

A. `2`
B. `gt 2`
C. `lt 2`
D. None of the above

Answer

To solve the expression \(\{(1 - \sin^2 \theta) \sec^2 \theta + \tan^2 \theta\} (\cos^2 \theta + 1)\), we can follow these steps:

### Step 1: Simplify \(1 - \sin^2 \theta\)
Using the Pythagorean identity, we know that:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 2: Substitute into the expression
Now, substitute \(1 - \sin^2 \theta\) into the expression:
\[ \{(1 - \sin^2 \theta) \sec^2 \theta + \tan^2 \theta\} = \{\cos^2 \theta \sec^2 \theta + \tan^2 \theta\} \]

### Step 3: Simplify \(\cos^2 \theta \sec^2 \theta\)
Recall that \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\), thus:
\[ \cos^2 \theta \sec^2 \theta = \cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1 \]
So the expression becomes:
\[ 1 + \tan^2 \theta \]

### Step 4: Use the identity \(1 + \tan^2 \theta\)
From the trigonometric identity, we know:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
Thus, we can rewrite the expression as:
\[ \sec^2 \theta \]

### Step 5: Multiply by \((\cos^2 \theta + 1)\)
Now we need to multiply this result by \((\cos^2 \theta + 1)\):
\[ \sec^2 \theta (\cos^2 \theta + 1) \]

### Step 6: Substitute \(\sec^2 \theta\)
Substituting \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\):
\[ \frac{1}{\cos^2 \theta} (\cos^2 \theta + 1) = \frac{\cos^2 \theta + 1}{\cos^2 \theta} \]

### Step 7: Simplify the expression
This can be simplified to:
\[ \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{1}{\cos^2 \theta} = 1 + \sec^2 \theta \]

### Final Expression
Thus, the final value of the expression is:
\[ 1 + \sec^2 \theta \]