Solution of the differential equation
`(dy)/(dx) + y sec x = tan x` is
`y(sec x +tan x) = sec x +tan x - x +c`.

Answer

To solve the differential equation

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

we will follow the steps for solving a first-order linear differential equation.

### Step 1: Identify \( p(x) \) and \( q(x) \)

In the given equation, we can identify:
- \( p(x) = \sec x \)
- \( q(x) = \tan x \)

### Step 2: Find the Integrating Factor

The integrating factor \( \mu(x) \) is given by the formula:

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

The integral of \( \sec x \) is \( \ln |\sec x + \tan x| \), so:

\[ \mu(x) = e^{\ln |\sec x + \tan x|} = \sec x + \tan x \]

### Step 3: Multiply the Differential Equation by the Integrating Factor

Now, we multiply the entire differential equation by the integrating factor:

\[ (\sec x + \tan x) \frac{dy}{dx} + y (\sec x + \tan x) \sec x = \tan x (\sec x + \tan x) \]

This simplifies to:

\[ (\sec x + \tan x) \frac{dy}{dx} + y \sec x (\sec x + \tan x) = \tan x (\sec x + \tan x) \]

### Step 4: Rewrite the Left Side as a Derivative

The left side can be rewritten as:

\[ \frac{d}{dx} \left[ y (\sec x + \tan x) \right] \]

Thus, we have:

\[ \frac{d}{dx} \left[ y (\sec x + \tan x) \right] = \tan x (\sec x + \tan x) \]

### Step 5: Integrate Both Sides

Now, we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx} \left[ y (\sec x + \tan x) \right] \, dx = \int \tan x (\sec x + \tan x) \, dx \]

The left side simplifies to:

\[ y (\sec x + \tan x) = \int \tan x (\sec x + \tan x) \, dx \]

### Step 6: Solve the Right Side Integral

To solve the right side, we can expand:

\[ \tan x (\sec x + \tan x) = \tan x \sec x + \tan^2 x \]

Now, we integrate:

1. The integral of \( \tan x \sec x \) is \( \sec x \).
2. The integral of \( \tan^2 x \) can be rewritten using the identity \( \tan^2 x = \sec^2 x - 1 \):

\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \tan x - x \]

Thus, we have:

\[ \int \tan x (\sec x + \tan x) \, dx = \sec x + \tan x - x + C \]

### Step 7: Combine Results

Putting everything together, we have:

\[ y (\sec x + \tan x) = \sec x + \tan x - x + C \]

### Step 8: Solve for \( y \)

Finally, we solve for \( y \):

\[ y = \frac{\sec x + \tan x - x + C}{\sec x + \tan x} \]

### Conclusion

Thus, the solution of the differential equation is:

\[ y(\sec x + \tan x) = \sec x + \tan x - x + C \]