The simplified value of `(SecA-cosA)^(2)+("cosec" A-sinA)^(2)-(cotA-tanA)^(2)` is

Options

A. 0
B. `(1)/(2)`
C. 1
D. 2

Answer

To simplify the expression \( (SecA - \cos A)^2 + (\csc A - \sin A)^2 - (\cot A - \tan A)^2 \), we can follow these steps:

### Step 1: Substitute Trigonometric Identities
We know the following trigonometric identities:
- \( \sec A = \frac{1}{\cos A} \)
- \( \csc A = \frac{1}{\sin A} \)
- \( \cot A = \frac{\cos A}{\sin A} \)
- \( \tan A = \frac{\sin A}{\cos A} \)

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

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

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

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

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

### Step 5: Further Simplification
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we can simplify the expression further.

### Step 6: Evaluate at \( A = 45^\circ \)
To find a specific value, we can substitute \( A = 45^\circ \):
- \( \sec 45^\circ = \sqrt{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \csc 45^\circ = \sqrt{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cot 45^\circ = 1 \)
- \( \tan 45^\circ = 1 \)

Substituting these values into the expression:
\[ (\sqrt{2} - \frac{1}{\sqrt{2}})^2 + (\sqrt{2} - \frac{1}{\sqrt{2}})^2 - (1 - 1)^2 \]

### Step 7: Calculate Each Term
Calculating each term:
1. First term:
\[ \left(\sqrt{2} - \frac{1}{\sqrt{2}}\right)^2 = \left(\frac{2 - 1}{\sqrt{2}}\right)^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
2. Second term:
\[ \left(\sqrt{2} - \frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
3. Third term:
\[ (1 - 1)^2 = 0 \]

### Step 8: Final Calculation
Combining these results:
\[ \frac{1}{2} + \frac{1}{2} - 0 = 1 \]

Thus, the simplified value of the expression is:
\[ \boxed{1} \]