Answer:
[tex] \text{To solve} \ \sin x + \sin 3x + \sin 7x + \sin 9x = 0, [/tex]
1. [tex] \text{Apply sum-to-product identities:} [/tex]
[tex] \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) [/tex]
2. [tex] \text{Use these identities to simplify the equation:} [/tex]
[tex] \sin x + \sin 9x = 2 \sin 5x \cos 4x [/tex]
[tex] \sin 3x + \sin 7x = 2 \sin 5x \cos 2x [/tex]
[tex] \text{Thus, the equation becomes:} [/tex]
[tex] 2 \sin 5x (\cos 4x + \cos 2x) = 0 [/tex]
3. [tex] \text{Solve for} \ \sin 5x = 0 \ \text{or} \ \cos 4x + \cos 2x = 0: [/tex]
- [tex] \sin 5x = 0 \ \text{gives} \ 5x = n\pi, \ \text{where} \ n \ \text{is an integer.} [/tex]
- [tex] \cos 4x + \cos 2x = 0 \ \text{requires further analysis using trigonometric identities or graphical methods.} [/tex]