Let a solution `y=y(x)` of the differential equation `(dy)/(dx)cosx+y sin x=1` satisfy y(0)=1
Statement-1: `y(x)=sin((pi)/(4)+x)`
Statement-2: The integrating factor of the given differential equation is sec x.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the differential equation \(\frac{dy}{dx} \cos x + y \sin x = 1\) with the initial condition \(y(0) = 1\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \frac{dy}{dx} \cos x + y \sin x = 1 \]
We can divide the entire equation by \(\cos x\) to simplify it:
\[ \frac{dy}{dx} + y \frac{\sin x}{\cos x} = \frac{1}{\cos x} \]
This simplifies to:
\[ \frac{dy}{dx} + y \tan x = \sec x \]

### Step 2: Identify \(p\) and \(q\)
This is now in the standard linear form:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
where \(p(x) = \tan x\) and \(q(x) = \sec x\).

### Step 3: Find the integrating factor
The integrating factor \(\mu(x)\) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \tan x \, dx} \]
The integral of \(\tan x\) is:
\[ \int \tan x \, dx = -\ln(\cos x) = \ln(\sec x) \]
Thus, the integrating factor is:
\[ \mu(x) = e^{\ln(\sec x)} = \sec x \]

### Step 4: Multiply the equation by the integrating factor
Now we multiply the entire differential equation by \(\sec x\):
\[ \sec x \frac{dy}{dx} + y \sec x \tan x = 1 \]
This can be rewritten as:
\[ \frac{d}{dx}(y \sec x) = 1 \]

### Step 5: Integrate both sides
Integrating both sides with respect to \(x\):
\[ y \sec x = \int 1 \, dx = x + C \]
Thus, we have:
\[ y \sec x = x + C \]
or
\[ y = (x + C) \cos x \]

### Step 6: Apply the initial condition
We use the initial condition \(y(0) = 1\):
\[ 1 = (0 + C) \cos(0) \]
Since \(\cos(0) = 1\), we get:
\[ 1 = C \]
Thus, \(C = 1\).

### Step 7: Write the final solution
Substituting \(C\) back into the equation:
\[ y = (x + 1) \cos x \]

### Step 8: Verify the statements
1. **Statement 1**: \(y(x) = \sin\left(\frac{\pi}{4} + x\right)\) is incorrect. The correct form we derived is \(y = (x + 1) \cos x\).
2. **Statement 2**: The integrating factor of the given differential equation is indeed \(\sec x\), which is correct.

### Conclusion
- **Statement 1** is false.
- **Statement 2** is true.