What is the value of `(sin^4alpha-cos^4 alpha+1)cosec^2 alpha`.

Options

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

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ (\sin^4 \alpha - \cos^4 \alpha + 1) \csc^2 \alpha \]
We know that \(\csc^2 \alpha = \frac{1}{\sin^2 \alpha}\), so we can rewrite the expression as:
\[ (\sin^4 \alpha - \cos^4 \alpha + 1) \cdot \frac{1}{\sin^2 \alpha} \]

### Step 2: Factor the difference of squares
Next, we recognize that \(\sin^4 \alpha - \cos^4 \alpha\) can be factored using the difference of squares:
\[ \sin^4 \alpha - \cos^4 \alpha = (\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha) \]
Since \(\sin^2 \alpha + \cos^2 \alpha = 1\), we can simplify this to:
\[ \sin^4 \alpha - \cos^4 \alpha = (\sin^2 \alpha - \cos^2 \alpha)(1) = \sin^2 \alpha - \cos^2 \alpha \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our expression:
\[ (\sin^2 \alpha - \cos^2 \alpha + 1) \cdot \frac{1}{\sin^2 \alpha} \]
This simplifies to:
\[ \frac{\sin^2 \alpha - \cos^2 \alpha + 1}{\sin^2 \alpha} \]

### Step 4: Simplify the numerator
Now, we can simplify the numerator:
\[ \sin^2 \alpha - \cos^2 \alpha + 1 = \sin^2 \alpha + 1 - \cos^2 \alpha = \sin^2 \alpha + 1 - (1 - \sin^2 \alpha) = 2\sin^2 \alpha \]
Thus, we have:
\[ \frac{2\sin^2 \alpha}{\sin^2 \alpha} = 2 \]

### Final Answer
Therefore, the value of the expression \((\sin^4 \alpha - \cos^4 \alpha + 1) \csc^2 \alpha\) is:
\[ \boxed{2} \]