1. **State the problem:** Simplify the expression $$\frac{\sin x \cdot \cos x}{\cos n x}$$ where $x$ and $n$ are variables. 2. **Recall relevant formulas:** There is no direct simplification unless we know the relationship between $n$ and $1$. However, we can express the numerator using a double-angle identity: $$\sin x \cos x = \frac{1}{2} \sin(2x)$$ 3. **Rewrite the expression:** $$\frac{\sin x \cos x}{\cos n x} = \frac{\frac{1}{2} \sin(2x)}{\cos n x} = \frac{1}{2} \cdot \frac{\sin(2x)}{\cos n x}$$ 4. **Interpretation:** Without additional information about $n$, this is the simplest form. If $n=1$, then: $$\frac{\sin x \cos x}{\cos x} = \sin x \cdot \cancel{\frac{\cos x}{\cos x}} = \sin x$$ 5. **Final answer:** $$\boxed{\frac{1}{2} \cdot \frac{\sin(2x)}{\cos n x}}$$ This is the simplified form unless $n=1$, in which case it simplifies further to $\sin x$.