1. **State the problem:** Solve the equation $3 + 2\cos^2(\theta) \tan(\theta) = 4 \cos^2(\theta)$ for $0^\circ \leq \theta \leq 360^\circ$. 2. **Recall the definitions and formulas:** - $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. - The equation involves trigonometric functions cosine and tangent. 3. **Rewrite the equation using $\tan(\theta)$ definition:** $$3 + 2 \cos^2(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)} = 4 \cos^2(\theta)$$ 4. **Simplify the left side:** $$3 + 2 \cos(\theta) \sin(\theta) = 4 \cos^2(\theta)$$ 5. **Bring all terms to one side:** $$3 + 2 \cos(\theta) \sin(\theta) - 4 \cos^2(\theta) = 0$$ 6. **Use the double-angle identity for sine:** Recall $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$, so $$2 \cos(\theta) \sin(\theta) = \sin(2\theta)$$ 7. **Rewrite the equation:** $$3 + \sin(2\theta) - 4 \cos^2(\theta) = 0$$ 8. **Express $\cos^2(\theta)$ using double-angle identity:** $$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$$ 9. **Substitute into the equation:** $$3 + \sin(2\theta) - 4 \cdot \frac{1 + \cos(2\theta)}{2} = 0$$ 10. **Simplify:** $$3 + \sin(2\theta) - 2 - 2 \cos(2\theta) = 0$$ 11. **Combine constants:** $$1 + \sin(2\theta) - 2 \cos(2\theta) = 0$$ 12. **Rewrite:** $$\sin(2\theta) - 2 \cos(2\theta) = -1$$ 13. **Let $x = 2\theta$, rewrite:** $$\sin x - 2 \cos x = -1$$ 14. **Solve for $x$:** Use the method of expressing $a \sin x + b \cos x = R \sin(x + \alpha)$ where $$R = \sqrt{1^2 + (-2)^2} = \sqrt{5}$$ and $$\alpha = \arctan\left(\frac{-2}{1}\right) = \arctan(-2)$$ 15. **Rewrite equation:** $$\sqrt{5} \sin(x + \alpha) = -1$$ 16. **Divide both sides:** $$\sin(x + \alpha) = \frac{-1}{\sqrt{5}}$$ 17. **Find general solutions:** $$x + \alpha = \arcsin\left(\frac{-1}{\sqrt{5}}\right) + 360^\circ k \quad \text{or} \quad 180^\circ - \arcsin\left(\frac{-1}{\sqrt{5}}\right) + 360^\circ k$$ 18. **Calculate $\alpha$ and $\arcsin$ values:** $$\alpha = \arctan(-2) \approx -63.435^\circ$$ $$\arcsin\left(\frac{-1}{\sqrt{5}}\right) \approx -26.565^\circ$$ 19. **Find $x$ values:** $$x = -26.565^\circ - (-63.435^\circ) + 360^\circ k = 36.87^\circ + 360^\circ k$$ $$x = 180^\circ - (-26.565^\circ) - (-63.435^\circ) + 360^\circ k = 270^\circ + 360^\circ k$$ 20. **Recall $x = 2\theta$, solve for $\theta$:** $$\theta = \frac{x}{2}$$ 21. **Find solutions for $0^\circ \leq \theta \leq 360^\circ$:** - For $x = 36.87^\circ + 360^\circ k$: - $k=0$: $\theta = 18.435^\circ$ - $k=1$: $\theta = 198.435^\circ$ - For $x = 270^\circ + 360^\circ k$: - $k=0$: $\theta = 135^\circ$ - $k=1$: $\theta = 315^\circ$ 22. **Final solutions:** $$\boxed{\theta = 18.435^\circ, 135^\circ, 198.435^\circ, 315^\circ}$$