The value of (cosecA+cotA+1) (cosec A-cot A+1) -2 cosec A is:

Options

A. 2
B. 0
C. 4 cosec A
D. 2 cosecA

Answer

To solve the expression \((\csc A + \cot A + 1)(\csc A - \cot A + 1) - 2 \csc A\), we will follow these steps:

### Step 1: Expand the expression
We start by expanding the product \((\csc A + \cot A + 1)(\csc A - \cot A + 1)\).

Using the distributive property (also known as the FOIL method for binomials):

\[ (\csc A + \cot A + 1)(\csc A - \cot A + 1) = \csc A(\csc A - \cot A + 1) + \cot A(\csc A - \cot A + 1) + 1(\csc A - \cot A + 1) \]

Calculating each term:
1. \(\csc A(\csc A) = \csc^2 A\)
2. \(\csc A(-\cot A) = -\csc A \cot A\)
3. \(\csc A(1) = \csc A\)

So, the first part becomes:
\[ \csc^2 A - \csc A \cot A + \csc A \]

Now for the second part:
1. \(\cot A(\csc A) = \cot A \csc A\)
2. \(\cot A(-\cot A) = -\cot^2 A\)
3. \(\cot A(1) = \cot A\)

So, the second part becomes:
\[ \cot A \csc A - \cot^2 A + \cot A \]

Finally, for the last part:
\[ 1(\csc A - \cot A + 1) = \csc A - \cot A + 1 \]

Combining all these results:
\[ \csc^2 A - \csc A \cot A + \csc A + \cot A \csc A - \cot^2 A + \cot A + \csc A - \cot A + 1 \]

### Step 2: Combine like terms
Now, we can combine all the terms:
- The \(\csc A\) terms: \(2\csc A\)
- The \(\cot A\) terms: \(-\cot^2 A + \cot A - \cot A = -\cot^2 A\)
- The \(-\csc A \cot A\) and \(+\cot A \csc A\) cancel each other out.
- The constant term is \(+1\).

Thus, we have:
\[ \csc^2 A - \cot^2 A + 2\csc A + 1 \]

### Step 3: Substitute back into the original expression
Now, we need to subtract \(2 \csc A\) from this result:
\[ (\csc^2 A - \cot^2 A + 2\csc A + 1) - 2\csc A \]
This simplifies to:
\[ \csc^2 A - \cot^2 A + 1 \]

### Step 4: Use the identity
Recall the identity:
\[ \csc^2 A - \cot^2 A = 1 \]
Thus, we can substitute:
\[ 1 + 1 = 2 \]

### Final Answer
The value of the expression is:
\[ \boxed{2} \]