If sin x + cosec x = 2. then what is the value of `sin^(6) x+ cosec^6x` ?

Options

A. 2
B. 4
C. 8
D. 16

Answer

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

### Step 1: Understand the given equation
We start with the equation:
\[ \sin x + \csc x = 2 \]
Recall that \( \csc x = \frac{1}{\sin x} \). Therefore, we can rewrite the equation as:
\[ \sin x + \frac{1}{\sin x} = 2 \]

### Step 2: Let \( y = \sin x \)
Let \( y = \sin x \). Then the equation becomes:
\[ y + \frac{1}{y} = 2 \]

### Step 3: Multiply through by \( y \)
To eliminate the fraction, multiply both sides by \( y \):
\[ y^2 + 1 = 2y \]

### Step 4: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ y^2 - 2y + 1 = 0 \]

### Step 5: Factor the quadratic
This can be factored as:
\[ (y - 1)^2 = 0 \]

### Step 6: Solve for \( y \)
Setting the factored equation to zero gives:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]
Thus, we find:
\[ \sin x = 1 \]

### Step 7: Find \( \csc x \)
Since \( \csc x = \frac{1}{\sin x} \), we have:
\[ \csc x = \frac{1}{1} = 1 \]

### Step 8: Calculate \( \sin^6 x + \csc^6 x \)
Now we can calculate \( \sin^6 x + \csc^6 x \):
\[ \sin^6 x + \csc^6 x = 1^6 + 1^6 = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( \sin^6 x + \csc^6 x \) is:
\[ \boxed{2} \]