If `sin x + sin y = a, cos x + cos y =b`, then
`sin ( x+y ) ="……….."`

Answer

To solve the problem, we start with the given equations:

1. \( \sin x + \sin y = a \)
2. \( \cos x + \cos y = b \)

We need to find \( \sin(x+y) \).

### Step 1: Use the sum-to-product identities
We can express \( \sin x + \sin y \) and \( \cos x + \cos y \) using the sum-to-product identities:

\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

From the first equation, we have:

\[ a = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

From the second equation, we have:

\[ b = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

### Step 2: Divide the two equations
Now, we can divide the two equations to eliminate \( \cos\left(\frac{x-y}{2}\right) \):

\[ \frac{a}{b} = \frac{2 \sin\left(\frac{x+y}{2}\right)}{2 \cos\left(\frac{x+y}{2}\right)} = \tan\left(\frac{x+y}{2}\right) \]

### Step 3: Express \( \sin(x+y) \)
Using the identity \( \sin(x+y) = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x+y}{2}\right) \), we can express \( \sin(x+y) \) in terms of \( a \) and \( b \):

\[ \sin(x+y) = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x+y}{2}\right) \]

### Step 4: Find \( \sin(x+y) \) in terms of \( a \) and \( b \)
We can express \( \sin(x+y) \) using the values of \( a \) and \( b \):

1. From \( a = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \), we can express \( \sin\left(\frac{x+y}{2}\right) \) as:
\[ \sin\left(\frac{x+y}{2}\right) = \frac{a}{2 \cos\left(\frac{x-y}{2}\right)} \]

2. From \( b = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \), we can express \( \cos\left(\frac{x+y}{2}\right) \) as:
\[ \cos\left(\frac{x+y}{2}\right) = \frac{b}{2 \cos\left(\frac{x-y}{2}\right)} \]

Now substituting these into the sine formula:

\[ \sin(x+y) = 2 \left(\frac{a}{2 \cos\left(\frac{x-y}{2}\right)}\right) \left(\frac{b}{2 \cos\left(\frac{x-y}{2}\right)}\right) \]

\[ = \frac{ab}{\cos^2\left(\frac{x-y}{2}\right)} \]

### Step 5: Use \( \cos^2\left(\frac{x-y}{2}\right) \)
From the identity \( \cos^2\left(\frac{x-y}{2}\right) = \frac{1 + \cos(x-y)}{2} \), we can express \( \sin(x+y) \) as:

\[ \sin(x+y) = \frac{2ab}{a^2 + b^2} \]

### Final Result
Thus, we conclude that:

\[ \sin(x+y) = \frac{2ab}{a^2 + b^2} \]