Step 1: Rewrite the differential equation in standard linear form.

The given differential equation is $\cos^2 x \frac{dy}{dx} + y = \tan x$. To convert this into the form $\frac{dy}{dx} + Py = Q$, we divide both sides by $\cos^2 x$ (valid since $0 \leq x < \frac{\pi}{2}$ implies $\cos x \neq 0$).

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

Here, $P = \sec^2 x$ and $Q = \tan x \sec^2 x$.

Step 2: Find the Integrating Factor (I.F.).

We calculate the integrating factor using the formula $e^{\int P \, dx}$.

$$\text{I.F.} = e^{\int \sec^2 x \, dx} = e^{\tan x}$$

Step 3: Apply the general solution formula.

The general solution for a linear differential equation is given by $y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.}) \, dx + C$.

Substituting the values we found:

$$y \cdot e^{\tan x} = \int (\tan x \sec^2 x) \cdot e^{\tan x} \, dx + C$$

Step 4: Evaluate the integral on the right-hand side.

Let $t = \tan x$. Then, differentiating, we get $dt = \sec^2 x \, dx$.

Substituting these into the integral:

$$\int t \cdot e^t \, dt$$

Using integration by parts, we get:

$$\int t \cdot e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t = e^t(t - 1)$$

Substituting $t = \tan x$ back into the expression:

$$\int (\tan x \sec^2 x) e^{\tan x} \, dx = e^{\tan x}(\tan x - 1)$$

Step 5: Write the final general solution.

Substituting the evaluated integral back into the solution equation:

$$y e^{\tan x} = e^{\tan x}(\tan x - 1) + C$$

Dividing both sides by $e^{\tan x}$, we get the required general solution:

$$y = \tan x - 1 + C e^{-\tan x}$$

∴ The general solution is $y = \tan x - 1 + C e^{-\tan x}$.