1. The problem is to simplify the expression $\tan x \cdot \sec x$. 2. Recall the definitions: $\tan x = \frac{\sin x}{\cos x}$ and $\sec x = \frac{1}{\cos x}$. 3. Substitute these into the expression: $$\tan x \cdot \sec x = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}$$ 4. Multiply the fractions: $$\frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = \frac{\sin x}{\cos^2 x}$$ 5. The simplified form is: $$\boxed{\frac{\sin x}{\cos^2 x}}$$ This is the final simplified expression for $\tan x \cdot \sec x$.