If `sin q + cosecq =2,` then the vlaue of `sin ^(6) q + cosec ^(6) q` is equal to

Options

A. 0
B. 1
C. 2
D. `2 ^(3)`

Answer

To solve the problem, we need to find the value of \( \sin^6 q + \csc^6 q \) given that \( \sin q + \csc q = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sin q + \csc q = 2 \]

2. **Rewrite \( \csc q \):**
\[ \csc q = \frac{1}{\sin q} \]
Thus, we can rewrite the equation as:
\[ \sin q + \frac{1}{\sin q} = 2 \]

3. **Multiply through by \( \sin q \) to eliminate the fraction:**
\[ \sin^2 q + 1 = 2 \sin q \]

4. **Rearrange the equation:**
\[ \sin^2 q - 2 \sin q + 1 = 0 \]

5. **Recognize this as a perfect square:**
\[ (\sin q - 1)^2 = 0 \]

6. **Solve for \( \sin q \):**
\[ \sin q - 1 = 0 \implies \sin q = 1 \]

7. **Find \( \csc q \):**
Since \( \sin q = 1 \), we have:
\[ \csc q = \frac{1}{\sin q} = \frac{1}{1} = 1 \]

8. **Calculate \( \sin^6 q + \csc^6 q \):**
\[ \sin^6 q + \csc^6 q = 1^6 + 1^6 = 1 + 1 = 2 \]

### Final Answer:
\[ \sin^6 q + \csc^6 q = 2 \]