If `sin^(2)alpha=cos^(2)alpha` , then the value of `(cot^(6)alpha-cot^(2)alpha)` is

Options

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

Answer

To solve the problem, we start with the given equation:

1. **Given**: \( \sin^2 \alpha = \cos^2 \alpha \)

2. **Using the Pythagorean Identity**: We know that \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Since \( \sin^2 \alpha = \cos^2 \alpha \), we can denote \( \sin^2 \alpha = x \). Therefore, we have:
\[ x + x = 1 \implies 2x = 1 \implies x = \frac{1}{2} \]
Thus, \( \sin^2 \alpha = \cos^2 \alpha = \frac{1}{2} \).

3. **Finding \( \cot^2 \alpha \)**: We know that:
\[ \cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]

4. **Calculating \( \cot^6 \alpha - \cot^2 \alpha \)**: Now we can substitute \( \cot^2 \alpha \) into the expression:
\[ \cot^6 \alpha = (\cot^2 \alpha)^3 = 1^3 = 1 \]
Therefore, we have:
\[ \cot^6 \alpha - \cot^2 \alpha = 1 - 1 = 0 \]

5. **Final Answer**: The value of \( \cot^6 \alpha - \cot^2 \alpha \) is \( 0 \).