1. **Problem (a):** Prove the identity $$\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta$$ by expressing $$3\theta$$ as $$2\theta + \theta$$. 2. Use the cosine addition formula: $$\cos(a + b) = \cos a \cos b - \sin a \sin b$$. 3. Substitute $$a = 2\theta$$ and $$b = \theta$$: $$\cos 3\theta = \cos(2\theta + \theta) = \cos 2\theta \cos \theta - \sin 2\theta \sin \theta$$. 4. Use double-angle formulas: $$\cos 2\theta = 2 \cos^2 \theta - 1$$ $$\sin 2\theta = 2 \sin \theta \cos \theta$$. 5. Substitute these into the expression: $$\cos 3\theta = (2 \cos^2 \theta - 1) \cos \theta - 2 \sin \theta \cos \theta \sin \theta$$. 6. Simplify the second term: $$2 \sin \theta \cos \theta \sin \theta = 2 \cos \theta \sin^2 \theta$$. 7. So, $$\cos 3\theta = 2 \cos^3 \theta - \cos \theta - 2 \cos \theta \sin^2 \theta$$. 8. Use $$\sin^2 \theta = 1 - \cos^2 \theta$$: $$\cos 3\theta = 2 \cos^3 \theta - \cos \theta - 2 \cos \theta (1 - \cos^2 \theta)$$. 9. Expand: $$\cos 3\theta = 2 \cos^3 \theta - \cos \theta - 2 \cos \theta + 2 \cos^3 \theta$$. 10. Combine like terms: $$\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$$. --- 11. **Problem (b):** Solve $$\cos 3\theta + \cos \theta \cos 2\theta = \cos^2 \theta$$ for $$0^\circ \leq \theta \leq 180^\circ$$. 12. Use the identity from (a): $$\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$$. 13. Also, $$\cos 2\theta = 2 \cos^2 \theta - 1$$. 14. Substitute into the equation: $$4 \cos^3 \theta - 3 \cos \theta + \cos \theta (2 \cos^2 \theta - 1) = \cos^2 \theta$$. 15. Expand: $$4 \cos^3 \theta - 3 \cos \theta + 2 \cos^3 \theta - \cos \theta = \cos^2 \theta$$. 16. Combine like terms: $$6 \cos^3 \theta - 4 \cos \theta = \cos^2 \theta$$. 17. Rearrange: $$6 \cos^3 \theta - \cos^2 \theta - 4 \cos \theta = 0$$. 18. Let $$x = \cos \theta$$, then: $$6 x^3 - x^2 - 4 x = 0$$. 19. Factor out $$x$$: $$x (6 x^2 - x - 4) = 0$$. 20. So, either $$x = 0$$ or solve quadratic: $$6 x^2 - x - 4 = 0$$. 21. Use quadratic formula: $$x = \frac{1 \pm \sqrt{1 + 96}}{12} = \frac{1 \pm \sqrt{97}}{12}$$. 22. Approximate roots: $$x_1 \approx 0.88, \quad x_2 \approx -0.76$$. 23. Find $$\theta$$ for each root in $$0^\circ \leq \theta \leq 180^\circ$$: - For $$x=0$$: $$\cos \theta = 0 \Rightarrow \theta = 90^\circ$$. - For $$x \approx 0.88$$: $$\theta \approx \cos^{-1}(0.88) = 28.4^\circ$$. - For $$x \approx -0.76$$: $$\theta \approx \cos^{-1}(-0.76) = 139.5^\circ$$. **Final solutions:** $$\theta = 28.4^\circ, 90^\circ, 139.5^\circ$$.