Президентский ФМЛ №239

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$\DeclareMathOperator{\tg}{tg}$ $\DeclareMathOperator{\ctg}{ctg}$

1. $\sin{2x} = 2\sin{x}\cos{x}$
2. Три формулы $\cos{2x}=\cos^2{x}-\sin^2{x}$
3. $\sin{3x}=3\sin{x}-4\sin^3{x}$
4. $\cos{3x}=4\cos^3{x}-3\cos{x}$
5. $\tg{2x}=\dfrac{2\tg{x}}{1-\tg^2{x}}$
6. $\sin{(x+y)}=\sin{x}\cos{y}+\cos{x}\sin{y}$
7. $\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}$
8. $\cos{(x+y)}=\cos{x}\cos{y}-\sin{x}\sin{y}$
9. $\cos{(x-y)}=\cos{x}\cos{y}+\sin{x}\sin{y}$
10. $\tg{(x+y)}=\dfrac{\tg{x}+\tg{y}}{1-\tg{x}\tg{y}}$
11. $\tg{(x-y)}=\dfrac{\tg{x}-\tg{y}}{1+\tg{x}\tg{y}}$
12. $\sin^2{x}=\dfrac{1-\cos{2x}}{2}$
13. $\cos^2{x}=\dfrac{1+\cos{2x}}{2}$
14. Универсальная подстановка $\sin{x}=\dfrac{2\tg{\dfrac{x}{2}}}{ 1+\tg^2{\dfrac{x}{2}}}$
15. Универсальная подстановка $\cos{x}=\dfrac{1-\tg{\dfrac{x}{2}}}{1+\tg^2{\dfrac{x}{2}}}$
16. $\cos{x}+\cos{y}=2\cos{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$
17. $\cos{x}-\cos{y}=-2\sin{\dfrac{x+y}{2}}\sin{\dfrac{x-y}{2}}$
18. $\sin{x}+\sin{y}=2\sin{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$
19. $\sin{x}-\sin{y}=2\sin{\dfrac{x-y}{2}}\cos{\dfrac{x+y}{2}}$
20. $\cos{x}\cos{y}=\dfrac{1}{2}(\cos{(x-y)}+\cos{(x+y)})$
21. $\sin{x}\sin{y}=\dfrac{1}{2}(\cos{(x-y)}-\cos{(x+y)})$
22. $\sin{x}\cos{y}=\dfrac{1}{2}(\sin{(x-y)}+\sin{(x+y)})$
23. $\dfrac{1}{\cos^2{x}}=\tg^2{x}+1$
24. $\dfrac{1}{\sin^2{x}}=\ctg^2{x}+1$
25. $\sin{(\pi-x)}$
26. $\sin{(\pi+x)}$
27. $\sin{\left(\dfrac{\pi}{2}-x\right)}$
28. $\sin{\left(\dfrac{\pi}{2}+x\right)}$
29. $\sin{\left(\dfrac{3\pi}{2}-x\right)}$
30. $\sin{\left(\dfrac{3\pi}{2}+x\right)}$
31. $\sin{(-x)}$
32. $\cos{(\pi-x)}$
33. $\cos{(\pi+x)}$
34. $\cos{\left(\dfrac{\pi}{2}-x\right)}$
35. $\cos{\left(\dfrac{\pi}{2}+x\right)}$
36. $\cos{\left(\dfrac{3\pi}{2}-x\right)}$
37. $\cos{\left(\dfrac{3\pi}{2}+x\right)}$
38. $\cos{(-x)}$
39. $\tg{(\pi-x)}$
40. $\tg{(\pi+x)}$
41. $\tg{\left(\dfrac{\pi}{2}-x\right)}$
42. $\tg{\left(\dfrac{\pi}{2}+x\right)}$
43. $\tg{\left(\dfrac{3\pi}{2}-x\right)}$
44. $\tg{\left(\dfrac{3\pi}{2}+x\right)}$
45. $\tg{(-x)}$
46. $\ctg{(\pi-x)}$
47. $\ctg{(\pi+x)}$
48. $\ctg{\left(\dfrac{\pi}{2}-x\right)}$
49. $\ctg{\left(\dfrac{\pi}{2}+x\right)}$
50. $\ctg{\left(\dfrac{3\pi}{2}-x\right)}$
51. $\ctg{\left(\dfrac{3\pi}{2}+x\right)}$
52. $\ctg{(-x)}$