If `sin theta + cosec theta = 2`, then the value of `sin^7theta + cosec^7 theta` is

Options

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

Answer

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

1. **Given:**
\[ \sin \theta + \csc \theta = 2 \]

2. **Understanding the terms:**
- Recall that \(\csc \theta = \frac{1}{\sin \theta}\).
- Therefore, we can rewrite the equation as:
\[ \sin \theta + \frac{1}{\sin \theta} = 2 \]

3. **Let \(x = \sin \theta\):**
- The equation now becomes:
\[ x + \frac{1}{x} = 2 \]

4. **Multiply through by \(x\):**
- To eliminate the fraction, multiply both sides by \(x\):
\[ x^2 + 1 = 2x \]

5. **Rearranging the equation:**
- Rearranging gives us a quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

6. **Factoring the quadratic:**
- This can be factored as:
\[ (x - 1)^2 = 0 \]

7. **Finding the value of \(x\):**
- Solving this gives:
\[ x - 1 = 0 \implies x = 1 \]
- Thus, \(\sin \theta = 1\).

8. **Finding \(\csc \theta\):**
- Since \(\csc \theta = \frac{1}{\sin \theta}\), we have:
\[ \csc \theta = \frac{1}{1} = 1 \]

9. **Calculating \( \sin^7 \theta + \csc^7 \theta \):**
- Now, we need to find:
\[ \sin^7 \theta + \csc^7 \theta = 1^7 + 1^7 = 1 + 1 = 2 \]

10. **Final Answer:**
- Therefore, the value of \(\sin^7 \theta + \csc^7 \theta\) is:
\[ \boxed{2} \]