1SIN(2A) - SIN(2A)*COS(2A)
= 2SIN(A)COS(A) - 2SIN(A)COS(A)*COS(2A)
= 2SIN(A)COS(A) - 2SIN(A)COS(A)*(1 - 2SIN^2(A))
= 2SIN(A)COS(A) - 2SIN(A)COS(A) + 4SIN^3(A)COS(A)
= 4SIN^3(A)COS(A)
= SIN(4A)
Таким образом, 1) доказано.
2SIN(A) / (1 + COS(A)) + CTG(A) = 1 / SIN(A)
= (SIN(A) / (1 + COS(A))) + (COS(A) / SIN(A))
= (SIN^2(A) + COS(A)) / SIN(A)
= (1 + COS(A)) / SIN(A)
= 1 / SIN(A)
Таким образом, 2) доказано.
31 - 2SIN^2(A) / (2COS^2(A) - 1)
= (2COS^2(A) - 1 - 2SIN^2(A)) / (2COS^2(A) - 1)
= (COS(2A) - COS(2A)) / (2COS^2(A) - 1)
= 0 / (2COS^2(A) - 1)
= 0
Таким образом, 3) упрощено до 0.
1
SIN(2A) - SIN(2A)*COS(2A)
= 2SIN(A)COS(A) - 2SIN(A)COS(A)*COS(2A)
= 2SIN(A)COS(A) - 2SIN(A)COS(A)*(1 - 2SIN^2(A))
= 2SIN(A)COS(A) - 2SIN(A)COS(A) + 4SIN^3(A)COS(A)
= 4SIN^3(A)COS(A)
= SIN(4A)
Таким образом, 1) доказано.
2
SIN(A) / (1 + COS(A)) + CTG(A) = 1 / SIN(A)
= (SIN(A) / (1 + COS(A))) + (COS(A) / SIN(A))
= (SIN^2(A) + COS(A)) / SIN(A)
= (1 + COS(A)) / SIN(A)
= 1 / SIN(A)
Таким образом, 2) доказано.
3
1 - 2SIN^2(A) / (2COS^2(A) - 1)
= (2COS^2(A) - 1 - 2SIN^2(A)) / (2COS^2(A) - 1)
= (COS(2A) - COS(2A)) / (2COS^2(A) - 1)
= 0 / (2COS^2(A) - 1)
= 0
Таким образом, 3) упрощено до 0.