The value of `((sintheta+sinphi)/(costheta+cosphi)+(costheta-cosphi)/(sintheta-sinphi))` 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 follow these steps:

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions. The common denominator will be \((\cos \theta + \cos \phi)(\sin \theta - \sin \phi)\).

\[ \text{Combined expression} = \frac{(\sin \theta + \sin \phi)(\sin \theta - \sin \phi) + (\cos \theta - \cos \phi)(\cos \theta + \cos \phi)}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} \]

### Step 2: Simplify the numerator
Now we simplify the numerator:

1. The first part:
\[ (\sin \theta + \sin \phi)(\sin \theta - \sin \phi) = \sin^2 \theta - \sin^2 \phi \]

2. The second part:
\[ (\cos \theta - \cos \phi)(\cos \theta + \cos \phi) = \cos^2 \theta - \cos^2 \phi \]

Combining these results, we get:
\[ \sin^2 \theta - \sin^2 \phi + \cos^2 \theta - \cos^2 \phi \]

### Step 3: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\) and \(\sin^2 \phi + \cos^2 \phi = 1\):
\[ \sin^2 \theta + \cos^2 \theta = 1 \quad \text{and} \quad \sin^2 \phi + \cos^2 \phi = 1 \]

Thus, we can rewrite the numerator:
\[ (1 - 1) = 0 \]

### Step 4: Final expression
Now substituting back into the combined expression:
\[ \frac{0}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} = 0 \]

### Conclusion
The value of the given expression is:
\[ \boxed{0} \]