The value of `((sin theta + sin phi)/(cos theta + cos phi)+ (cos theta - cos phi)/(sin theta - sin phi))` is :

Options

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

Answer

To solve the expression \(\frac{\sin \theta + \sin \phi}{\cos \theta + \cos \phi} + \frac{\cos \theta - \cos \phi}{\sin \theta - \sin \phi}\), we will break it down step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\sin \theta + \sin \phi}{\cos \theta + \cos \phi} + \frac{\cos \theta - \cos \phi}{\sin \theta - \sin \phi} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((\cos \theta + \cos \phi)(\sin \theta - \sin \phi)\). We rewrite the expression:
\[ \frac{(\sin \theta + \sin \phi)(\sin \theta - \sin \phi) + (\cos \theta - \cos \phi)(\cos \theta + \cos \phi)}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} \]

### Step 3: Expand the numerator
Now we expand the numerator:
1. For the first term:
\[ (\sin \theta + \sin \phi)(\sin \theta - \sin \phi) = \sin^2 \theta - \sin^2 \phi \]
2. For the second term:
\[ (\cos \theta - \cos \phi)(\cos \theta + \cos \phi) = \cos^2 \theta - \cos^2 \phi \]

So, the numerator becomes:
\[ \sin^2 \theta - \sin^2 \phi + \cos^2 \theta - \cos^2 \phi \]

### Step 4: Use the Pythagorean identity
Using the identity \(\sin^2 x + \cos^2 x = 1\), we can simplify:
\[ (\sin^2 \theta + \cos^2 \theta) - (\sin^2 \phi + \cos^2 \phi) = 1 - 1 = 0 \]

### Step 5: Substitute back into the expression
Now substituting back into the expression, we have:
\[ \frac{0}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} = 0 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{0} \]