I see you've uploaded an image with trigonometric problems. Let me help you solve them step-by-step. We'll begin with question 1.

Question 1

Rewrite each expression as a single trigonometric ratio.

(a) $\sin a \cos 2a + \cos a \sin 2a$

Using the sum formula for sine:
$\sin x \cos y + \cos x \sin y = \sin(x + y)$ $\sin a \cos 2a + \cos a \sin 2a = \sin(a + 2a) = \sin 3a$

(b) $\cos 4x \cos 3x - \sin 4x \sin 3x$

Using the cosine subtraction formula:
$\cos x \cos y - \sin x \sin y = \cos(x + y)$ $\cos 4x \cos 3x - \sin 4x \sin 3x = \cos(4x + 3x) = \cos 7x$

Would you like me to continue solving Question 2, or provide more explanation for Question 1? Let me know!
Here are some follow-up questions for additional learning:

1. How does the sine sum formula differ from the cosine sum formula?
2. What are the exact values of $\sin 30^\circ$ and $\cos 45^\circ$?
3. Can these formulas be applied to angles outside $0^\circ$ to $360^\circ$?
4. How would you derive these compound angle formulas geometrically?
5. Why is it important to simplify trigonometric expressions?

Tip: Memorizing fundamental trigonometric formulas helps speed up problem-solving in complex expressions.