To simplify the expression $\frac{2 \sin(x)}{1} \cdot \frac{\sin(x)}{\cos(x)}$, let’s go through each part step by step:

1. Simplify the first part: $\frac{2 \sin(x)}{1}$ is simply $2 \sin(x)$.

2. Substitute the expression, so we have: $2 \sin(x) \cdot \frac{\sin(x)}{\cos(x)}$

3. Combine terms by multiplying: $\frac{2 \sin(x) \cdot \sin(x)}{\cos(x)} = \frac{2 \sin^2(x)}{\cos(x)}$

4. Now, recall that $\sin^2(x)$ can be written in terms of $\tan(x)$ because $\frac{\sin^2(x)}{\cos(x)} = \sin(x) \cdot \tan(x)$.

5. This gives us the simplified expression: $2 \sin(x) \tan(x)$

So, the simplified form is: $2 \sin(x) \tan(x)$

Would you like more details on trigonometric simplifications? Here are some questions that may help deepen your understanding:

1. How can we further simplify expressions using trigonometric identities?
2. What are some common uses of the identity $\sin^2(x) + \cos^2(x) = 1$?
3. How does the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$ assist in simplifying expressions?
4. What are other equivalent forms of $\sin(x) \tan(x)$?
5. How can we verify that two trigonometric expressions are equal?

Tip: Remember, using trigonometric identities can simplify complex expressions quickly, making it easier to work with equations involving trig functions.