If y= y(x) is the solution of the differential equation
`dy/dx=(tan x-y) sec^(2)x, x in(-pi/2,pi/2),` such that y (0)=0,
than `y(-pi/4)` is equal to

Options

A. e-2
B. `2+1/e`
C. `1/e-2`
D. `1/2-e`

Explanation

To solve the differential equation given by

\[ \frac{dy}{dx} = (tan x - y) \sec^2 x \]

with the initial condition \( y(0) = 0 \), we will follow these steps:

### Step 1: Rewrite the differential equation
We can rewrite the equation as:

\[ \frac{dy}{dx} + y \sec^2 x = \tan x \sec^2 x \]

### Step 2: Identify the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int \sec^2 x \, dx} = e^{\tan x} \]

### Step 3: Multiply the entire equation by the integrating factor
Multiplying the entire differential equation by \( e^{\tan x} \):

\[ e^{\tan x} \frac{dy}{dx} + y e^{\tan x} \sec^2 x = \tan x e^{\tan x} \sec^2 x \]

### Step 4: Recognize the left side as a derivative
The left-hand side can be recognized as the derivative of a product:

\[ \frac{d}{dx}(y e^{\tan x}) = \tan x e^{\tan x} \sec^2 x \]

### Step 5: Integrate both sides
Integrating both sides with respect to \( x \):

\[ y e^{\tan x} = \int \tan x e^{\tan x} \sec^2 x \, dx \]

Using the substitution \( t = \tan x \), \( dt = \sec^2 x \, dx \):

\[ y e^{\tan x} = \int t e^t \, dt \]

### Step 6: Solve the integral using integration by parts
Using integration by parts where \( u = t \) and \( dv = e^t dt \):

\[ \int t e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t + C \]

Substituting back:

\[ y e^{\tan x} = \tan x e^{\tan x} - e^{\tan x} + C \]

### Step 7: Solve for \( y \)
Dividing through by \( e^{\tan x} \):

\[ y = \tan x - 1 + Ce^{-\tan x} \]

### Step 8: Apply the initial condition
Using the initial condition \( y(0) = 0 \):

\[ 0 = \tan(0) - 1 + Ce^{-\tan(0)} \implies 0 = 0 - 1 + C \implies C = 1 \]

### Step 9: Substitute \( C \) back into the equation
Thus, the solution becomes:

\[ y = \tan x - 1 + e^{-\tan x} \]

### Step 10: Find \( y(-\frac{\pi}{4}) \)
Now, we need to evaluate \( y(-\frac{\pi}{4}) \):

\[ y\left(-\frac{\pi}{4}\right) = \tan\left(-\frac{\pi}{4}\right) - 1 + e^{-\tan\left(-\frac{\pi}{4}\right)} \]

Calculating \( \tan\left(-\frac{\pi}{4}\right) = -1 \):

\[ y\left(-\frac{\pi}{4}\right) = -1 - 1 + e^{1} = -2 + e \]

### Final Answer
Thus,

\[ y\left(-\frac{\pi}{4}\right) = e - 2 \]