1)
a)Cos(α-β) - Cos(α+β) = CosαCosβ + SinαSinβ -(CosαCosβ-SinαSinβ)=
=CosαCosβ + SinαSinβ -CosαCosβ +SinαSinβ =
=2SinαSinβ.
б)числитель = Sin2α = 2SinαCosα
знаменатель = 2Sinα
Ответ: Сosα
в) 2Sin5αCos3α -Sin8α = 2Sin5αCos3α -Sin(5α+3α) =
=2Sin5αCos3α -Sin5αCos3α - Cos5αSin3α = Sin5αCos3α -Cos5αSin3α=
=Sin(5α-3α) = Sin2α.
2)
а)Sin4x = -√2/2
4x= (-1)^n arcSin(-√2/2) + nπ, n∈Z
4x = (-1)^n*(-π/4) + nπ, n ∈Z
4x = (-1)^(n+1)*π/4 + nπ, n ∈Z
x = (-1)^(n+1)*π/16 + nπ/4, n ∈Z
б)3Cos²x + 7Sinx -5 = 0
3(1 -Sin²x) + 7Sinx -5=0
3 -3Sin²x + 7Sinx -5 = 0
-3Sin²x + 7Sinx -2 = 0
Sinx = t
3t² -7t +2 = 0
D = b²-4ac = 49 - 24 = 25
t₁ = 2 t₂ = 1/3
Sinx = 2 Sinx = 1/3
∅ x = (-1)^narcSin(1/3) + nπ, n ∈ Z
3)
а) Cosx < 1/2
-π/3 +2πk < x < π/3 + 2πk , k ∈Z
б) tgx > -√3/3
-π/6 + πk < x < π/2 + πk , k ∈Z
,