If sin (x+y)=cos (x-y), then the value of `cos^2x` is :

Options

A. 3
B. 5
C. `1/4`
D. `1/2`

Answer

To solve the equation \( \sin(x+y) = \cos(x-y) \) and find the value of \( \cos^2 x \), we can follow these steps:

### Step 1: Use the sine and cosine addition formulas
We know that:
\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]
\[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]
So we can rewrite the equation as:
\[ \sin x \cos y + \cos x \sin y = \cos x \cos y + \sin x \sin y \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ \sin x \cos y + \cos x \sin y - \cos x \cos y - \sin x \sin y = 0 \]
This simplifies to:
\[ \sin x \cos y - \sin x \sin y + \cos x \sin y - \cos x \cos y = 0 \]

### Step 3: Factor the equation
We can factor by grouping:
\[ \sin x (\cos y - \sin y) + \cos x (\sin y - \cos y) = 0 \]
This can be rewritten as:
\[ \sin x (\cos y - \sin y) + \cos x (-\cos y + \sin y) = 0 \]

### Step 4: Set up the factors
From this, we can set up two possible equations:
1. \( \sin x = 0 \) or \( \cos y - \sin y = 0 \)
2. \( \cos x = 0 \) or \( \sin y - \cos y = 0 \)

### Step 5: Solve for \( x \)
Assuming \( \sin x = \cos x \), we have:
\[ \tan x = 1 \]
This implies:
\[ x = \tan^{-1}(1) = 45^\circ \]

### Step 6: Calculate \( \cos^2 x \)
Now, we find \( \cos^2 x \):
\[ \cos^2 x = \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \cos^2 x \) is:
\[ \boxed{\frac{1}{2}} \]