1. **State the problem:** Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written as $$5 \cos^2 \theta - \cos \theta - 4 = 0$$. 2. **Rewrite the given equation:** Start with $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$. Recall that $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$. 3. **Substitute $$\tan \theta$$:** $$\cos \theta - 1 = 4 \sin \theta \times \frac{\sin \theta}{\cos \theta} = \frac{4 \sin^2 \theta}{\cos \theta}$$. 4. **Multiply both sides by $$\cos \theta$$ to clear the denominator:** $$\cancel{\cos \theta} (\cos \theta - 1) \cancel{\cos \theta} = 4 \sin^2 \theta$$ which gives $$\cos^2 \theta - \cos \theta = 4 \sin^2 \theta$$. 5. **Use the Pythagorean identity $$\sin^2 \theta = 1 - \cos^2 \theta$$:** $$\cos^2 \theta - \cos \theta = 4 (1 - \cos^2 \theta)$$. 6. **Expand the right side:** $$\cos^2 \theta - \cos \theta = 4 - 4 \cos^2 \theta$$. 7. **Bring all terms to one side:** $$\cos^2 \theta - \cos \theta - 4 + 4 \cos^2 \theta = 0$$ which simplifies to $$5 \cos^2 \theta - \cos \theta - 4 = 0$$. This proves part (a). --- **Part (b): Solve for $$0 \leq x < \frac{\pi}{2}$$ the equation:** $$\cos 2x - 1 = 4 \sin 2x \tan 2x$$. 1. Recognize this is the same form as part (a) with $$\theta = 2x$$. 2. From part (a), the equivalent quadratic is: $$5 \cos^2 (2x) - \cos (2x) - 4 = 0$$. 3. Let $$y = \cos (2x)$$, then solve: $$5 y^2 - y - 4 = 0$$. 4. Use the quadratic formula: $$y = \frac{1 \pm \sqrt{(-1)^2 - 4 \times 5 \times (-4)}}{2 \times 5} = \frac{1 \pm \sqrt{1 + 80}}{10} = \frac{1 \pm \sqrt{81}}{10} = \frac{1 \pm 9}{10}$$. 5. Two solutions for $$y$$: - $$y = \frac{1 + 9}{10} = 1$$ - $$y = \frac{1 - 9}{10} = -\frac{8}{10} = -0.8$$ 6. Solve for $$x$$: - If $$\cos (2x) = 1$$, then $$2x = 0$$ (within $$0 \leq 2x < \pi$$), so $$x = 0$$. - If $$\cos (2x) = -0.8$$, then $$2x = \cos^{-1}(-0.8)$$. Calculate $$\cos^{-1}(-0.8) \approx 2.4981$$ radians. 7. Since $$0 \leq x < \frac{\pi}{2}$$, then $$0 \leq 2x < \pi$$, so only one solution for $$2x$$ in $$[0, \pi)$$ is $$2.4981$$. 8. Therefore, $$x = \frac{2.4981}{2} = 1.24905 \approx 1.25$$ radians. **Final answers:** $$x = 0$$ and $$x \approx 1.25$$ radians (to 2 decimal places).