The value of `cosycos(pi/2-x)-cos(pi/2-y)cosx+sinycos(pi/2-x)+cosxsin(pi/2-y)` is zero if `x=0` (b) `y=0` `x=y` (d) `npi+y-pi/4(n in Z)`

Options

A. `x=0`
B. `y=0`
C. `x=y`
D. `npi+y-pi/4 (n in Z)`

Explanation

To solve the given expression:

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

we will simplify it step by step.

### Step 1: Use Trigonometric Identities

Recall the following trigonometric identities:
- \(\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta\)
- \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta\)

Using these identities, we can rewrite the expression:

\[ \cos y \sin x - \sin y \cos x + \sin y \sin x + \cos x \cos y \]

### Step 2: Group Terms

Now, we can group the terms:

\[ (\cos y \sin x + \sin y \sin x) + (\cos x \cos y - \sin y \cos x) \]

This can be factored as:

\[ \sin x (\cos y + \sin y) + \cos x (\cos y - \sin y) \]

### Step 3: Set the Expression to Zero

We want to find when this expression is equal to zero:

\[ \sin x (\cos y + \sin y) + \cos x (\cos y - \sin y) = 0 \]

### Step 4: Solve for Conditions

This equation can be satisfied in two cases:

1. \(\sin x = 0\) or \(\cos x = 0\)
2. The coefficients must balance each other.

From the first case:
- If \(\sin x = 0\), then \(x = n\pi\) where \(n \in \mathbb{Z}\).
- If \(\cos x = 0\), then \(x = \frac{\pi}{2} + n\pi\).

From the second case, we can set up the equation:

\[ \frac{\sin x}{\cos x} = \frac{\cos y - \sin y}{\cos y + \sin y} \]

This leads us to the condition:

\[ \tan x = \frac{\cos y - \sin y}{\cos y + \sin y} \]

### Step 5: Analyze the Conditions

To find the conditions on \(y\):
- If \(x = y\), then both sides are equal.
- If \(y = 0\), we can find specific values of \(x\) that satisfy the equation.

### Conclusion

After analyzing the conditions, we find that the expression is zero if:

\[ x = n\pi + y - \frac{\pi}{4}, \quad n \in \mathbb{Z} \]

Thus, the correct option is:

**(d)** \(n\pi + y - \frac{\pi}{4}, \, n \in \mathbb{Z}\)