Evaluate for `sin^(29) x + cosec^(29)x`, if `sin x + cosec x=2`.

Options

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

Answer

To solve the problem, we need to evaluate \( \sin^{29} x + \csc^{29} x \) given that \( \sin x + \csc x = 2 \).

### Step-by-Step Solution:

1. **Understanding the Given Equation:**
We start with the equation:
\[ \sin x + \csc x = 2 \]
We know that \( \csc x = \frac{1}{\sin x} \). Therefore, we can rewrite the equation as:
\[ \sin x + \frac{1}{\sin x} = 2 \]

2. **Multiplying Through by \( \sin x \):**
To eliminate the fraction, we multiply both sides by \( \sin x \):
\[ \sin^2 x + 1 = 2 \sin x \]

3. **Rearranging the Equation:**
Rearranging the equation gives us:
\[ \sin^2 x - 2 \sin x + 1 = 0 \]

4. **Factoring the Quadratic:**
The left-hand side can be factored as:
\[ (\sin x - 1)^2 = 0 \]
This implies:
\[ \sin x - 1 = 0 \quad \Rightarrow \quad \sin x = 1 \]

5. **Finding \( \csc x \):**
Since \( \sin x = 1 \), we can find \( \csc x \):
\[ \csc x = \frac{1}{\sin x} = \frac{1}{1} = 1 \]

6. **Calculating \( \sin^{29} x + \csc^{29} x \):**
Now we substitute \( \sin x \) and \( \csc x \) into the expression we need to evaluate:
\[ \sin^{29} x + \csc^{29} x = 1^{29} + 1^{29} = 1 + 1 = 2 \]

### Final Answer:
\[ \sin^{29} x + \csc^{29} x = 2 \]