Expression `(sin^(4) x - cos^(4)x +1)"cosec"^(2)x` is equal to:

Options

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

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ (\sin^4 x - \cos^4 x + 1) \csc^2 x \]
We can rewrite \(\sin^4 x\) and \(\cos^4 x\) using the identity for squares:
\[ \sin^4 x = (\sin^2 x)^2 \quad \text{and} \quad \cos^4 x = (\cos^2 x)^2 \]
Thus, we have:
\[ (\sin^2 x)^2 - (\cos^2 x)^2 + 1 \]

### Step 2: Apply the difference of squares
The expression \(\sin^4 x - \cos^4 x\) can be factored using the difference of squares:
\[ a^2 - b^2 = (a + b)(a - b) \]
Letting \(a = \sin^2 x\) and \(b = \cos^2 x\), we get:
\[ (\sin^2 x + \cos^2 x)(\sin^2 x - \cos^2 x) + 1 \]
Since \(\sin^2 x + \cos^2 x = 1\), we can simplify:
\[ 1(\sin^2 x - \cos^2 x) + 1 = \sin^2 x - \cos^2 x + 1 \]

### Step 3: Substitute and simplify
Now, we substitute this back into the expression:
\[ (\sin^2 x - \cos^2 x + 1) \csc^2 x \]
Using the identity \(\csc^2 x = \frac{1}{\sin^2 x}\), we can rewrite the expression:
\[ (\sin^2 x - \cos^2 x + 1) \cdot \frac{1}{\sin^2 x} \]

### Step 4: Distribute \(\csc^2 x\)
Distributing \(\csc^2 x\) gives:
\[ \frac{\sin^2 x}{\sin^2 x} - \frac{\cos^2 x}{\sin^2 x} + \frac{1}{\sin^2 x} \]
This simplifies to:
\[ 1 - \cot^2 x + \csc^2 x \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \csc^2 x = 1 + \cot^2 x \]
Substituting this into our expression gives:
\[ 1 - \cot^2 x + (1 + \cot^2 x) = 1 - \cot^2 x + 1 + \cot^2 x \]
The \(-\cot^2 x\) and \(+\cot^2 x\) cancel each other out, leaving us with:
\[ 2 \]

### Final Result
Thus, the final value of the expression \((\sin^4 x - \cos^4 x + 1) \csc^2 x\) is:
\[ \boxed{2} \]