If `sin theta+ cosec theta =2,` then the value of `sin^8 theta + cosec^8 theta` is equal to

Options

A. 2
B. `2^(4)`
C. `2^(8)`
D. more than `2^(8)`

Answer

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

\[ \sin \theta + \csc \theta = 2 \]

### Step 1: Rewrite \(\csc \theta\)
Recall that \(\csc \theta\) is the reciprocal of \(\sin \theta\). Therefore, we can rewrite the equation as:

\[ \sin \theta + \frac{1}{\sin \theta} = 2 \]

### Step 2: Multiply through by \(\sin \theta\)
To eliminate the fraction, multiply both sides of the equation by \(\sin \theta\):

\[ \sin^2 \theta + 1 = 2 \sin \theta \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:

\[ \sin^2 \theta - 2 \sin \theta + 1 = 0 \]

### Step 4: Factor the quadratic
The equation can be factored as:

\[ (\sin \theta - 1)^2 = 0 \]

### Step 5: Solve for \(\sin \theta\)
Setting the factor equal to zero gives:

\[ \sin \theta - 1 = 0 \implies \sin \theta = 1 \]

### Step 6: Find \(\csc \theta\)
Since \(\sin \theta = 1\), we find:

\[ \csc \theta = \frac{1}{\sin \theta} = 1 \]

### Step 7: Calculate \(\sin^8 \theta + \csc^8 \theta\)
Now we can compute \(\sin^8 \theta + \csc^8 \theta\):

\[ \sin^8 \theta + \csc^8 \theta = 1^8 + 1^8 = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \(\sin^8 \theta + \csc^8 \theta\) is:

\[ \boxed{2} \]