1. **State the problem:** Evaluate the definite integral $$\int_0^{\frac{\pi}{4}} \tan x \sec^2 x \, dx$$ using substitution. 2. **Recall the substitution method:** We look for a substitution $u = f(x)$ such that the integral simplifies. Here, notice that the derivative of $\tan x$ is $\sec^2 x$, which appears in the integral. 3. **Choose substitution:** Let $$u = \tan x$$ Then, $$\frac{du}{dx} = \sec^2 x \implies du = \sec^2 x \, dx$$ 4. **Rewrite the integral:** Substitute $\tan x = u$ and $\sec^2 x \, dx = du$: $$\int_0^{\frac{\pi}{4}} \tan x \sec^2 x \, dx = \int_{u=\tan 0}^{u=\tan \frac{\pi}{4}} u \, du$$ 5. **Evaluate the new limits:** $$\tan 0 = 0, \quad \tan \frac{\pi}{4} = 1$$ So the integral becomes: $$\int_0^1 u \, du$$ 6. **Integrate:** $$\int_0^1 u \, du = \left. \frac{u^2}{2} \right|_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}$$ 7. **Final answer:** $$\boxed{\frac{1}{2}}$$