If `o^(@)ltAlt90^(@)` , then the value of `tan^(2)A+cot^(2)A-sec^(2)A"cosec"^(2)A` is

Options

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

Answer

To solve the problem, we need to evaluate the expression \( \tan^2 A + \cot^2 A - \sec^2 A \cdot \csc^2 A \).

### Step-by-Step Solution:

1. **Write down the definitions of the trigonometric functions:**
- \( \tan A = \frac{\sin A}{\cos A} \)
- \( \cot A = \frac{\cos A}{\sin A} \)
- \( \sec A = \frac{1}{\cos A} \)
- \( \csc A = \frac{1}{\sin A} \)

2. **Substitute the definitions into the expression:**
\[ \tan^2 A + \cot^2 A - \sec^2 A \cdot \csc^2 A = \left(\frac{\sin^2 A}{\cos^2 A}\right) + \left(\frac{\cos^2 A}{\sin^2 A}\right) - \left(\frac{1}{\cos^2 A} \cdot \frac{1}{\sin^2 A}\right) \]

3. **Combine the terms:**
\[ = \frac{\sin^4 A + \cos^4 A}{\sin^2 A \cos^2 A} - \frac{1}{\sin^2 A \cos^2 A} \]
\[ = \frac{\sin^4 A + \cos^4 A - 1}{\sin^2 A \cos^2 A} \]

4. **Use the identity \( \sin^2 A + \cos^2 A = 1 \) to simplify \( \sin^4 A + \cos^4 A \):**
\[ \sin^4 A + \cos^4 A = (\sin^2 A + \cos^2 A)^2 - 2\sin^2 A \cos^2 A = 1 - 2\sin^2 A \cos^2 A \]

5. **Substitute this back into the expression:**
\[ = \frac{(1 - 2\sin^2 A \cos^2 A) - 1}{\sin^2 A \cos^2 A} \]
\[ = \frac{-2\sin^2 A \cos^2 A}{\sin^2 A \cos^2 A} \]

6. **Simplify the expression:**
\[ = -2 \]

### Final Answer:
The value of \( \tan^2 A + \cot^2 A - \sec^2 A \cdot \csc^2 A \) is \( -2 \).