If `sin theta+cosec theta=2`, then find the value of `sin^(20)theta+cosec^(20)theta`.

Answer

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

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

### Step 1: Rewrite \(\csc \theta\)
We know that \(\csc \theta\) is the reciprocal of \(\sin \theta\), so we can rewrite the equation as:

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

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

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

### Step 3: Rearrange the equation
Now, we rearrange the equation to form a standard quadratic equation:

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

### Step 4: Factor the quadratic
This can be factored as:

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

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

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

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

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

### Step 7: Calculate \(\sin^{20} \theta + \csc^{20} \theta\)
Now we need to find:

\[ \sin^{20} \theta + \csc^{20} \theta \]

Substituting the values we found:

\[ \sin^{20} \theta + \csc^{20} \theta = 1^{20} + 1^{20} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \(\sin^{20} \theta + \csc^{20} \theta\) is:

\[ \boxed{2} \]

---