The `sintheta+cosectheta=2,` then: `sin^2theta+cosec^2theta=`

Options

A. 1
B. 4
C. 2
D. 8

Answer

To solve the equation \( \sin \theta + \csc \theta = 2 \) and find the value of \( \sin^2 \theta + \csc^2 \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \csc \theta = \frac{1}{\sin \theta} \). Thus, 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 by \( \sin \theta \):
\[ \sin^2 \theta + 1 = 2 \sin \theta \]

### Step 3: Rearrange the equation
Rearranging gives us a 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:
\[ \sin \theta - 1 = 0 \implies \sin \theta = 1 \]

### 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 \( \sin^2 \theta + \csc^2 \theta \)
Now we can calculate \( \sin^2 \theta + \csc^2 \theta \):
\[ \sin^2 \theta + \csc^2 \theta = 1^2 + 1^2 = 1 + 1 = 2 \]

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