What is the value of (cosec A - sin A) (sec A - cos A) (tan A + cot A) ?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the value of the expression \((\csc A - \sin A)(\sec A - \cos A)(\tan A + \cot A)\), we will simplify each part step by step.

### Step 1: Rewrite in terms of sine and cosine
We start by rewriting the trigonometric functions in terms of sine and cosine:
- \(\csc A = \frac{1}{\sin A}\)
- \(\sec A = \frac{1}{\cos A}\)
- \(\tan A = \frac{\sin A}{\cos A}\)
- \(\cot A = \frac{\cos A}{\sin A}\)

So, we can rewrite the expression as:
\[ \left(\frac{1}{\sin A} - \sin A\right)\left(\frac{1}{\cos A} - \cos A\right)\left(\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\right) \]

### Step 2: Simplify each term
1. **First term**:
\[ \frac{1}{\sin A} - \sin A = \frac{1 - \sin^2 A}{\sin A} = \frac{\cos^2 A}{\sin A} \]
(using the identity \(1 - \sin^2 A = \cos^2 A\))

2. **Second term**:
\[ \frac{1}{\cos A} - \cos A = \frac{1 - \cos^2 A}{\cos A} = \frac{\sin^2 A}{\cos A} \]
(using the identity \(1 - \cos^2 A = \sin^2 A\))

3. **Third term**:
\[ \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \tan A + \cot A = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} = \frac{1}{\sin A \cos A} \]
(using the identity \(\sin^2 A + \cos^2 A = 1\))

### Step 3: Combine all terms
Now substituting back into the expression:
\[ \left(\frac{\cos^2 A}{\sin A}\right)\left(\frac{\sin^2 A}{\cos A}\right)\left(\frac{1}{\sin A \cos A}\right) \]

### Step 4: Simplify the entire expression
Combining these fractions:
\[ = \frac{\cos^2 A \cdot \sin^2 A \cdot 1}{\sin A \cdot \cos A \cdot \sin A \cdot \cos A} = \frac{\cos^2 A \sin^2 A}{\sin^2 A \cos^2 A} = 1 \]

### Final Answer
Thus, the value of the expression \((\csc A - \sin A)(\sec A - \cos A)(\tan A + \cot A)\) is:
\[ \boxed{1} \]