If `sinx=cosy` where `x` and y are acute angle then what is the relation between `x` and y ?

Options

A. `x-y=(pi)/(2)`
B. `x+y=(3pi)/(2)`
C. `x+y=(pi)/(2)`
D. `x+y=(pi)/(4)`

Answer

To solve the problem where \( \sin x = \cos y \) and \( x \) and \( y \) are acute angles, we can follow these steps:

### Step 1: Use the co-function identity
We know from trigonometric identities that:
\[ \sin x = \cos\left(\frac{\pi}{2} - x\right) \]
This means that if \( \sin x = \cos y \), we can express \( \cos y \) in terms of sine:
\[ \cos y = \sin\left(\frac{\pi}{2} - y\right) \]

### Step 2: Set the equations equal
Since \( \sin x = \cos y \), we can rewrite it using the co-function identity:
\[ \sin x = \sin\left(\frac{\pi}{2} - y\right) \]

### Step 3: Use the property of sine function
For two angles \( A \) and \( B \), if \( \sin A = \sin B \), then:
\[ A = B \quad \text{or} \quad A = \pi - B \]
Applying this to our equation:
\[ x = \frac{\pi}{2} - y \quad \text{(since both angles are acute)} \]

### Step 4: Rearranging the equation
From the equation \( x = \frac{\pi}{2} - y \), we can rearrange it:
\[ x + y = \frac{\pi}{2} \]

### Conclusion
Thus, the relation between \( x \) and \( y \) is:
\[ x + y = \frac{\pi}{2} \]