The value of `cos((pi)/(4)-theta)cos((pi)/(4)-phi)-sin((pi)/(4)-theta)sin((pi)/(4)-phi)` will be

Options

A. `sin(theta-phi)`
B. `sin(theta+phi)`
C. `cos(theta-phi)`
D. `cos(theta+phi)`

Answer

To solve the expression \( \cos\left(\frac{\pi}{4} - \theta\right) \cos\left(\frac{\pi}{4} - \phi\right) - \sin\left(\frac{\pi}{4} - \theta\right) \sin\left(\frac{\pi}{4} - \phi\right) \), we can use the cosine of the sum formula.

### Step-by-step Solution:

1. **Identify the Formula**: We will use the cosine of the sum formula, which states:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
In our case, we can set \( a = \frac{\pi}{4} - \theta \) and \( b = \frac{\pi}{4} - \phi \).

2. **Rewrite the Expression**: Using the formula, we can rewrite our expression:
\[ \cos\left(\frac{\pi}{4} - \theta\right) \cos\left(\frac{\pi}{4} - \phi\right) - \sin\left(\frac{\pi}{4} - \theta\right) \sin\left(\frac{\pi}{4} - \phi\right) = \cos\left(\left(\frac{\pi}{4} - \theta\right) + \left(\frac{\pi}{4} - \phi\right)\right) \]

3. **Simplify the Argument**: Now, simplify the argument of the cosine:
\[ \left(\frac{\pi}{4} - \theta\right) + \left(\frac{\pi}{4} - \phi\right) = \frac{\pi}{4} + \frac{\pi}{4} - \theta - \phi = \frac{\pi}{2} - \theta - \phi \]

4. **Final Expression**: Thus, we have:
\[ \cos\left(\frac{\pi}{2} - \theta - \phi\right) \]

5. **Use the Cosine Identity**: We know that:
\[ \cos\left(\frac{\pi}{2} - x\right) = \sin x \]
Therefore:
\[ \cos\left(\frac{\pi}{2} - \theta - \phi\right) = \sin(\theta + \phi) \]

### Conclusion:
The value of the expression \( \cos\left(\frac{\pi}{4} - \theta\right) \cos\left(\frac{\pi}{4} - \phi\right) - \sin\left(\frac{\pi}{4} - \theta\right) \sin\left(\frac{\pi}{4} - \phi\right) \) is:
\[ \sin(\theta + \phi) \]