`(1)/((1+sin x)(2+sin x))=(a)/(1+sin x )- (b)/(2+sin x)` then `a+b=`

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

To solve the equation

\[ \frac{1}{(1+\sin x)(2+\sin x)} = \frac{a}{1+\sin x} - \frac{b}{2+\sin x} \]

we will use the method of partial fractions. Let's break this down step by step.

### Step 1: Rewrite the equation
We start with the equation:

\[ \frac{1}{(1+\sin x)(2+\sin x)} = \frac{a}{1+\sin x} - \frac{b}{2+\sin x} \]

### Step 2: Find a common denominator
To combine the right-hand side, we need a common denominator, which is \((1+\sin x)(2+\sin x)\):

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

### Step 3: Set the numerators equal
Since the denominators are the same, we can set the numerators equal to each other:

\[ 1 = a(2+\sin x) - b(1+\sin x) \]

### Step 4: Expand the right-hand side
Expanding the right-hand side gives:

\[ 1 = 2a + a\sin x - b - b\sin x \]

### Step 5: Rearranging the equation
Rearranging the equation, we get:

\[ 1 = (2a - b) + (a - b)\sin x \]

### Step 6: Compare coefficients
For the equation to hold for all \(x\), the coefficients of \(\sin x\) must be equal on both sides. This gives us two equations:

1. Coefficient of \(\sin x\): \(a - b = 0\) (1)
2. Constant term: \(2a - b = 1\) (2)

### Step 7: Solve the system of equations
From equation (1), we have:

\[ a = b \]

Substituting \(a = b\) into equation (2):

\[ 2a - a = 1 \implies a = 1 \]

Since \(a = b\), we also have:

\[ b = 1 \]

### Step 8: Calculate \(a + b\)
Now, we can find \(a + b\):

\[ a + b = 1 + 1 = 2 \]

Thus, the final answer is:

\[ \boxed{2} \]