It looks like you've uploaded an image with some trigonometric questions. Here's what I can see:

1. The first part involves simplifying an expression: $\sin(2x) \cos(x) - \cos(2x) \sin(x)$

2. The second part asks to find the exact value of $\cos(x)$ given that: $\sin(x) = -\frac{5}{13}, \quad \pi \leq x \leq \frac{3\pi}{2}$

I'll start with solving these two problems. First, for the trigonometric simplification:

Simplifying $\sin(2x) \cos(x) - \cos(2x) \sin(x)$

We recognize that this expression is a variation of the sine addition formula: $\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)$ By applying the identity to the given expression, we have: $\sin(2x) \cos(x) - \cos(2x) \sin(x) = \sin(2x - x) = \sin(x)$ So, the simplified expression is $\sin(x)$.

Finding $\cos(x)$ given $\sin(x) = -\frac{5}{13}$

Using the identity: $\sin^2(x) + \cos^2(x) = 1$ We substitute $\sin(x) = -\frac{5}{13}$: $\left(-\frac{5}{13}\right)^2 + \cos^2(x) = 1$ $\frac{25}{169} + \cos^2(x) = 1$ $\cos^2(x) = 1 - \frac{25}{169} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169}$ $\cos(x) = \pm \frac{12}{13}$ Since $\pi \leq x \leq \frac{3\pi}{2}$ (which places $x$ in the third quadrant), we take the negative root: $\cos(x) = -\frac{12}{13}$

Would you like further clarification on any part of this solution? Here are some related questions you might find interesting:

1. How does the sine addition formula apply to other expressions?
2. What are the implications of using different trigonometric identities for simplifying expressions?
3. How can you solve for $\sin(x)$ when $\cos(x)$ is given in a certain quadrant?
4. What is the significance of quadrant placement when determining the signs of trigonometric functions?
5. How can you use these methods to solve more complex trigonometric identities?

Tip: Always check the quadrant to determine the correct sign for trigonometric values when working with inverse or known function values!