`(sinA+sinB)/(cosA-cosB)+(cosA+cosB)/(sinA-sinB)=?`

Options

A. sinAcosB
B. tanAtanB
C. 0
D. cosAcosB

Answer

To solve the expression \((\sin A + \sin B) / (\cos A - \cos B) + (\cos A + \cos B) / (\sin A - \sin B)\), we can follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\cos A - \cos B)(\sin A - \sin B)\).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{(\sin A + \sin B)(\sin A - \sin B) + (\cos A + \cos B)(\cos A - \cos B)}{(\cos A - \cos B)(\sin A - \sin B)} \]

### Step 3: Expand the numerators
Now we expand the numerators:
1. For \((\sin A + \sin B)(\sin A - \sin B)\):
\[ \sin^2 A - \sin^2 B \]
2. For \((\cos A + \cos B)(\cos A - \cos B)\):
\[ \cos^2 A - \cos^2 B \]

### Step 4: Combine the numerators
Now, we combine the results from the expansions:
\[ \sin^2 A - \sin^2 B + \cos^2 A - \cos^2 B \]

### Step 5: Use the Pythagorean identity
Using the identity \(\sin^2 A + \cos^2 A = 1\) and \(\sin^2 B + \cos^2 B = 1\), we can simplify:
\[ (\sin^2 A + \cos^2 A) - (\sin^2 B + \cos^2 B) = 1 - 1 = 0 \]

### Step 6: Final expression
Thus, the entire expression simplifies to:
\[ \frac{0}{(\cos A - \cos B)(\sin A - \sin B)} = 0 \]

### Final Answer:
The value of the expression is \(0\).

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