Let's break down the expression step-by-step, using one rule at a time.

The given expression is:

\frac{\frac{2 \frac{\sin x}{\cos x}}{\frac{\cos^2 x}{\cos x} + \frac{\sin^2 x}{\cos^2 x}}}

Step 1: Simplify Fractions in the Denominator

Observe that the denominator has terms of the form $\frac{\cos^2 x}{\cos x}$ and $\frac{\sin^2 x}{\cos^2 x}$. Let's simplify each of these fractions:

1. $\frac{\cos^2 x}{\cos x} = \cos x$
2. $\frac{\sin^2 x}{\cos^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x$

After this simplification, the expression becomes:

$\frac{\frac{2 \sin x}{\cos x}}{\cos x + \tan^2 x}$

Would you like to proceed with the next step?

Related Questions:

1. What is the next simplification rule to apply here?
2. How can we use trigonometric identities like Pythagorean identities to simplify further?
3. What happens if we rewrite $\tan^2 x$ in terms of $\sin x$ and $\cos x$?
4. Can we factor or simplify further after applying Pythagorean identities?
5. How does changing each trigonometric function to sine and cosine help simplify expressions?

Tip: Breaking complex trigonometric expressions down to simpler sine and cosine forms often reveals opportunities for simplification.