Evaluate the simplified value of `(1 + cot^2 theta) (1 - cos theta ) (1 + cos theta)`.

Options

A. 1
B. `-1`
C. `cot theta `
D. `sec^2 theta `

Answer

To evaluate the expression \((1 + \cot^2 \theta)(1 - \cos \theta)(1 + \cos \theta)\), we can follow these steps:

### Step 1: Rewrite \(1 + \cot^2 \theta\)
Using the identity \(1 + \cot^2 \theta = \csc^2 \theta\), we can rewrite the expression:
\[ (1 + \cot^2 \theta) = \csc^2 \theta \]
So, the expression becomes:
\[ \csc^2 \theta (1 - \cos \theta)(1 + \cos \theta) \]

### Step 2: Simplify \((1 - \cos \theta)(1 + \cos \theta)\)
Using the difference of squares formula, we have:
\[ (1 - \cos \theta)(1 + \cos \theta) = 1^2 - \cos^2 \theta = 1 - \cos^2 \theta \]

### Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know:
\[ 1 - \cos^2 \theta = \sin^2 \theta \]
Thus, we can substitute this back into our expression:
\[ \csc^2 \theta \cdot \sin^2 \theta \]

### Step 4: Substitute \(\csc^2 \theta\)
Recall that \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\). Therefore, we can rewrite the expression:
\[ \frac{1}{\sin^2 \theta} \cdot \sin^2 \theta \]

### Step 5: Simplify the expression
Now, when we multiply \(\frac{1}{\sin^2 \theta}\) by \(\sin^2 \theta\), the \(\sin^2 \theta\) terms cancel out:
\[ \frac{1 \cdot \sin^2 \theta}{\sin^2 \theta} = 1 \]

### Final Answer
Thus, the simplified value of the expression \((1 + \cot^2 \theta)(1 - \cos \theta)(1 + \cos \theta)\) is:
\[ \boxed{1} \]