`(cosec A - sin A)^2 + (sec A - cos A)^2 - (cot A - tan A)^2` is equal to

Options

A. 1
B. 2
C. 0
D. `-1`

Answer

To solve the expression \((\csc A - \sin A)^2 + (\sec A - \cos A)^2 - (\cot A - \tan A)^2\), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
Recall the definitions of the trigonometric functions:
- \(\csc A = \frac{1}{\sin A}\)
- \(\sec A = \frac{1}{\cos A}\)
- \(\cot A = \frac{\cos A}{\sin A}\)
- \(\tan A = \frac{\sin A}{\cos A}\)

### Step 2: Substitute the definitions into the expression
Substituting the definitions into the expression, we have:
\[ \left(\frac{1}{\sin A} - \sin A\right)^2 + \left(\frac{1}{\cos A} - \cos A\right)^2 - \left(\frac{\cos A}{\sin A} - \frac{\sin A}{\cos A}\right)^2 \]

### Step 3: Simplify each term
1. **First term**:
\[ \left(\frac{1 - \sin^2 A}{\sin A}\right)^2 = \left(\frac{\cos^2 A}{\sin A}\right)^2 = \frac{\cos^4 A}{\sin^2 A} \]

2. **Second term**:
\[ \left(\frac{1 - \cos^2 A}{\cos A}\right)^2 = \left(\frac{\sin^2 A}{\cos A}\right)^2 = \frac{\sin^4 A}{\cos^2 A} \]

3. **Third term**:
\[ \left(\frac{\cos^2 A - \sin^2 A}{\sin A \cos A}\right)^2 = \left(\frac{\cos^2 A - \sin^2 A}{\sin A \cos A}\right)^2 = \frac{(\cos^2 A - \sin^2 A)^2}{\sin^2 A \cos^2 A} \]

### Step 4: Combine the terms
Now we can combine the terms:
\[ \frac{\cos^4 A}{\sin^2 A} + \frac{\sin^4 A}{\cos^2 A} - \frac{(\cos^2 A - \sin^2 A)^2}{\sin^2 A \cos^2 A} \]

### Step 5: Find a common denominator
The common denominator for the first two terms is \(\sin^2 A \cos^2 A\):
\[ \frac{\cos^4 A \cos^2 A + \sin^4 A \sin^2 A - (\cos^2 A - \sin^2 A)^2}{\sin^2 A \cos^2 A} \]

### Step 6: Simplify the numerator
The numerator simplifies as follows:
- \(\cos^4 A + \sin^4 A - (\cos^2 A - \sin^2 A)^2 = \cos^4 A + \sin^4 A - (\cos^4 A - 2\cos^2 A \sin^2 A + \sin^4 A)\)
- This simplifies to \(2\cos^2 A \sin^2 A\).

### Step 7: Final expression
Thus, we have:
\[ \frac{2\cos^2 A \sin^2 A}{\sin^2 A \cos^2 A} = 2 \]

### Conclusion
The final result is:
\[ \boxed{2} \]