To solve the given differential equation, we can use the method of separating variables.

Separating variables involves isolating the terms with y and x on opposite sides of the equation.

The given differential equation is:

dy/dx - y tan(x) - y^2 sec(x) = 0

To begin, let's rearrange the equation:

dy/dx = y tan(x) + y^2 sec(x)

Now, we can separate the variables by moving all the terms involving y to one side and all the terms involving x to the other side:

dy/(y tan(x) + y^2 sec(x)) = dx

Next, we can integrate both sides of the equation.

Integrating the left side:

∫(1/(y tan(x) + y^2 sec(x))) dy

To simplify the integration, we can factor out y from the denominator:

∫(1/y(tan(x) + y sec(x))) dy

Now, we can use partial fraction decomposition to rewrite the integrand.

1/(y(tan(x) + y sec(x))) = A/y + B/(tan(x) + y sec(x))

Multiplying through by the denominator, we get:

1 = A(tan(x) + y sec(x)) + By

Expanding and equating coefficients, we find:

A = 1
B = -1

So, the integrand becomes:

1/y - 1/(tan(x) + y sec(x))

Now, we can integrate both terms separately:

∫(1/y) dy - ∫(1/(tan(x) + y sec(x))) dy

The integral of 1/y with respect to y is ln|y|.

For the second integral, we can use a substitution:

Let u = tan(x) + y sec(x)
Then, du = sec^2(x) + y sec(x) tan(x) dx

Rearranging, we have:

du - sec^2(x) dx = y sec(x) tan(x) dx
du - sec^2(x) dx = y du

Substituting back into the integral:

- ∫(1/u) du

This becomes -ln|u|.

Therefore, the integral of the second term is -ln|tan(x) + y sec(x)|.

Putting it all together, the integrated equation is:

ln|y| - ln|tan(x) + y sec(x)| = C

where C is the constant of integration.

Now, we can simplify the equation using logarithmic properties:

ln|y/(tan(x) + y sec(x))| = C

Exponentiating both sides:

|y/(tan(x) + y sec(x))| = e^C

Since e^C is a positive constant, we can remove the absolute value signs:

y/(tan(x) + y sec(x)) = e^C

Multiplying both sides by (tan(x) + y sec(x)), we get:

y = e^C (tan(x) + y sec(x))

Expanding, we have:

y = e^C tan(x) + e^C y sec(x)

Now, we can solve for y:

y - e^C y sec(x) = e^C tan(x