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I need the next step for the proof. 1 rule and 1 step at a time.

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.

Follow a step-by-step guide to simplify a trigonometric expression using trigonometric identities and algebraic techniques. This example covers Pythagorean identities, quotient identities, and careful fraction simplification. Suitable for high school students studying advanced trigonometry.

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