What is the value of `(sin^(4)theta-cos^(4)theta+1)"cosec"^(2)theta`?

Options

A. `-2`
B. 0
C. 1
D. 2

Explanation

To solve the expression \( \sin^4 \theta - \cos^4 \theta + 1 \cdot \csc^2 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin^4 \theta - \cos^4 \theta + 1 \cdot \csc^2 \theta \]

### Step 2: Use the difference of squares
The term \( \sin^4 \theta - \cos^4 \theta \) can be factored using the difference of squares:
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^2 \theta - \cos^2 \theta) \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we can simplify this to:
\[ \sin^4 \theta - \cos^4 \theta = 1 \cdot (\sin^2 \theta - \cos^2 \theta) = \sin^2 \theta - \cos^2 \theta \]

### Step 3: Substitute back into the expression
Now substituting back into the original expression gives us:
\[ \sin^2 \theta - \cos^2 \theta + \csc^2 \theta \]

### Step 4: Rewrite \(\csc^2 \theta\)
Recall that \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \). Thus, we can rewrite the expression as:
\[ \sin^2 \theta - \cos^2 \theta + \frac{1}{\sin^2 \theta} \]

### Step 5: Combine terms
To combine the terms, we can express everything in terms of \(\sin^2 \theta\):
\[ \sin^2 \theta - \cos^2 \theta + \frac{1}{\sin^2 \theta} = \sin^2 \theta - (1 - \sin^2 \theta) + \frac{1}{\sin^2 \theta} \]
This simplifies to:
\[ 2\sin^2 \theta - 1 + \frac{1}{\sin^2 \theta} \]

### Step 6: Final expression
Thus, the final expression is:
\[ 2\sin^2 \theta - 1 + \frac{1}{\sin^2 \theta} \]

### Conclusion
The value of the expression \( \sin^4 \theta - \cos^4 \theta + 1 \cdot \csc^2 \theta \) simplifies to:
\[ 2\sin^2 \theta - 1 + \frac{1}{\sin^2 \theta} \]