The value of `(1+ cot theta + "cosec"theta) (1+ cot theta - "cosec"theta)` is:

Options

A. `2tan theta`
B. `2 cot theta`
C. `2 sec theta`
D. `2"cosec"theta`

Answer

To solve the expression \((1 + \cot \theta + \csc \theta)(1 + \cot \theta - \csc \theta)\), we can follow these steps:

### Step 1: Recognize the Structure
The expression can be recognized as a product of two binomials, which can be simplified using the identity for the difference of squares:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, let \(A = 1 + \cot \theta\) and \(B = \csc \theta\). Thus, we rewrite the expression as:
\[ (A + B)(A - B) = A^2 - B^2 \]

### Step 2: Calculate \(A^2\)
Now, calculate \(A^2\):
\[ A^2 = (1 + \cot \theta)^2 = 1^2 + 2 \cdot 1 \cdot \cot \theta + \cot^2 \theta = 1 + 2 \cot \theta + \cot^2 \theta \]

### Step 3: Calculate \(B^2\)
Next, calculate \(B^2\):
\[ B^2 = \csc^2 \theta \]

### Step 4: Substitute Back into the Expression
Now substitute \(A^2\) and \(B^2\) back into the expression:
\[ A^2 - B^2 = (1 + 2 \cot \theta + \cot^2 \theta) - \csc^2 \theta \]

### Step 5: Use the Identity
Recall the identity:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
Substituting this into our expression gives:
\[ A^2 - B^2 = (1 + 2 \cot \theta + \cot^2 \theta) - (1 + \cot^2 \theta) \]

### Step 6: Simplify the Expression
Now simplify:
\[ A^2 - B^2 = 1 + 2 \cot \theta + \cot^2 \theta - 1 - \cot^2 \theta \]
The \(1\) and \(-1\) cancel out, and \(\cot^2 \theta\) also cancels out:
\[ A^2 - B^2 = 2 \cot \theta \]

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

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