If `(d^(2))/(d x ^(2))((sin ^(4)x+ sin ^(2)x+1)/(sin ^(2)x + si n x+1))= a sin ^(2) x+b sin x+c ` then the value of `b+c -a` is

Answer

To solve the problem, we need to find the second derivative of the function given and express it in the form \( a \sin^2 x + b \sin x + c \). Let's go through the steps one by one.

### Step 1: Simplify the Expression
We start with the expression:

\[ \frac{\sin^4 x + \sin^2 x + 1}{\sin^2 x + \sin x + 1} \]

Using the identity \( a^4 + a^2 + 1 = (a^2 + a + 1)(a^2 - a + 1) \), let \( a = \sin x \):

\[ \sin^4 x + \sin^2 x + 1 = (\sin^2 x + \sin x + 1)(\sin^2 x - \sin x + 1) \]

Thus, we can rewrite the expression as:

\[ \frac{(\sin^2 x + \sin x + 1)(\sin^2 x - \sin x + 1)}{\sin^2 x + \sin x + 1} \]

The \( \sin^2 x + \sin x + 1 \) terms cancel out, leaving us with:

\[ \sin^2 x - \sin x + 1 \]

### Step 2: Find the First Derivative
Let \( y = \sin^2 x - \sin x + 1 \). We now find the first derivative \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{d}{dx}(\sin^2 x) - \frac{d}{dx}(\sin x) + \frac{d}{dx}(1) \]

Using the chain rule and the derivative of sine:

\[ \frac{dy}{dx} = 2 \sin x \cos x - \cos x + 0 \]

This simplifies to:

\[ \frac{dy}{dx} = (2 \sin x - 1) \cos x \]

### Step 3: Find the Second Derivative
Now, we need to find the second derivative \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}((2 \sin x - 1) \cos x) \]

Using the product rule:

\[ \frac{d^2y}{dx^2} = (2 \cos x)(\cos x) + (2 \sin x - 1)(-\sin x) \]

This expands to:

\[ \frac{d^2y}{dx^2} = 2 \cos^2 x - (2 \sin^2 x - \sin x) \]

### Step 4: Simplify the Second Derivative
Using the identity \( \cos^2 x = 1 - \sin^2 x \):

\[ \frac{d^2y}{dx^2} = 2(1 - \sin^2 x) - (2 \sin^2 x - \sin x) \]

This simplifies to:

\[ \frac{d^2y}{dx^2} = 2 - 2 \sin^2 x - 2 \sin^2 x + \sin x \]

Combining like terms gives:

\[ \frac{d^2y}{dx^2} = 2 - 4 \sin^2 x + \sin x \]

### Step 5: Compare with the Given Form
We need to express this in the form \( a \sin^2 x + b \sin x + c \):

\[ \frac{d^2y}{dx^2} = -4 \sin^2 x + 1 \sin x + 2 \]

From this, we can identify:

- \( a = -4 \)
- \( b = 1 \)
- \( c = 2 \)

### Step 6: Calculate \( b + c - a \)
Now, we calculate:

\[ b + c - a = 1 + 2 - (-4) = 1 + 2 + 4 = 7 \]

### Final Answer
Thus, the value of \( b + c - a \) is:

\[ \boxed{7} \]