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Let's solve the two parts of the question by expressing each given trigonometric sum or difference as a product of sine and cosine functions.

### a) \( \cos(5x) + \cos(3x) \)

We can use the **sum-to-product identity** for cosines:

\[
\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)
\]

For \( A = 5x \) and \( B = 3x \):

\[
\cos(5x) + \cos(3x) = 2 \cos\left(\frac{5x + 3x}{2}\right) \cos\left(\frac{5x - 3x}{2}\right)
\]

Simplifying:

\[
= 2 \cos\left( \frac{8x}{2} \right) \cos\left( \frac{2x}{2} \right)
\]
\[
= 2 \cos(4x) \cos(x)
\]

Thus, the expression becomes:

\[
\cos(5x) + \cos(3x) = 2 \cos(4x) \cos(x)
\]

---

### b) \( \sin\left(\frac{x}{2}\right) - \sin\left(\frac{5x}{2}\right) \)

We can use the **difference-to-product identity** for sines:

\[
\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)
\]

For \( A = \frac{x}{2} \) and \( B = \frac{5x}{2} \):

\[
\sin\left(\frac{x}{2}\right) - \sin\left(\frac{5x}{2}\right) = 2 \cos\left(\frac{\frac{x}{2} + \frac{5x}{2}}{2}\right) \sin\left(\frac{\frac{x}{2} - \frac{5x}{2}}{2}\right)
\]

Simplifying:

\[
= 2 \cos\left(\frac{6x}{4}\right) \sin\left(\frac{-4x}{4}\right)
\]
\[
= 2 \cos\left(\frac{3x}{2}\right) \sin\left(-x\right)
\]
\[
= -2 \cos\left(\frac{3x}{2}\right) \sin(x)
\]

Thus, the expression becomes:

\[
\sin\left(\frac{x}{2}\right) - \sin\left(\frac{5x}{2}\right) = -2 \cos\left(\frac{3x}{2}\right) \sin(x)
\]

---

Would you like further details or clarification on any step?

Here are 5 related questions for practice:
1. Can you apply the sum-to-product identity for \( \sin(4x) + \sin(2x) \)?
2. How would you express \( \cos(7x) - \cos(5x) \) as a product?
3. What is the product form of \( \sin(6x) - \sin(2x) \)?
4. Can you convert \( \cos(9x) + \cos(7x) \) into a product of sines or cosines?
5. How does the identity for \( \sin A + \sin B \) differ from \( \cos A + \cos B \)?

**Tip**: Always remember to double-check your identities; some, like sum-to-product, work with specific angles for sine and cosine functions.

Write each expression as a product of sines and/or cosines: a) cos(5x) + cos(3x), b) sin(x/2) - sin(5x/2)

Learn how to express the trigonometric sums cos(5x) + cos(3x) and sin(x/2) - sin(5x/2) as products of sines and cosines using sum-to-product and difference-to-product identities. Detailed steps for high school trigonometry students in Grades 11-12.

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