4. Решите уравнение: 1) 2cos²3x - 1 = 0; 2) cos⁴5x - sin⁴5x = 0; 3) √3 cos(π/3 + x) + 3/2 sin x = √3/2; 4) 1 - 3sin²3x = sin 3x - 3(1 - cos 3x)(1 + cos 3x)

Решение:

1. \( 2\cos^2 3x - 1 = 0 \)


2. \( \cos^2 3x = \frac{1}{2} \)


3. \( \cos 3x = \pm \frac{1}{\sqrt{2}} \)


4. \( 3x = \frac{\pi}{4} + \frac{\pi k}{2} \)


5. \( x = \frac{\pi}{12} + \frac{\pi k}{6}, k \in \mathbb{Z} \)


6. 2) \( \cos^4 5x - \sin^4 5x = 0 \)


7. \( (\cos^2 5x - \sin^2 5x)(\cos^2 5x + \sin^2 5x) = 0 \)


8. \( \cos 10x \cdot 1 = 0 \)


9. \( 10x = \frac{\pi}{2} + \pi k \)


10. \( x = \frac{\pi}{20} + \frac{\pi k}{10}, k \in \mathbb{Z} \)


11. 3) \( \sqrt{3} \cos (\frac{\pi}{3} + x) + \frac{3}{2} \sin x = \frac{\sqrt{3}}{2} \)


12. \( \sqrt{3} (\cos \frac{\pi}{3} \cos x - \sin \frac{\pi}{3} \sin x) + \frac{3}{2} \sin x = \frac{\sqrt{3}}{2} \)


13. \( \sqrt{3} (\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x) + \frac{3}{2} \sin x = \frac{\sqrt{3}}{2} \)


14. \( \frac{\sqrt{3}}{2} \cos x - \frac{3}{2} \sin x + \frac{3}{2} \sin x = \frac{\sqrt{3}}{2} \)


15. \( \frac{\sqrt{3}}{2} \cos x = \frac{\sqrt{3}}{2} \)


16. \( \cos x = 1 \)


17. \( x = 2\pi k, k \in \mathbb{Z} \)


18. 4) \( 1 - 3\sin^2 3x = \sin 3x - 3(1 - \cos 3x)(1 + \cos 3x) \)


19. \( 1 - 3\sin^2 3x = \sin 3x - 3(1 - \cos^2 3x) \)


20. \( 1 - 3\sin^2 3x = \sin 3x - 3\sin^2 3x \)


21. \( 1 = \sin 3x \)


22. \( 3x = \frac{\pi}{2} + 2\pi k \)


23. \( x = \frac{\pi}{6} + \frac{2\pi k}{3}, k \in \mathbb{Z} \)

Ответ: 1) \( x = \frac{\pi}{12} + \frac{\pi k}{6}, k \in \mathbb{Z} \); 2) \( x = \frac{\pi}{20} + \frac{\pi k}{10}, k \in \mathbb{Z} \); 3) \( x = 2\pi k, k \in \mathbb{Z} \); 4) \( x = \frac{\pi}{6} + \frac{2\pi k}{3}, k \in \mathbb{Z} \).