x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x
x y : ℂ
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x
have h1 : x + 2 * x = 3 * x := by ring
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * cos (2 * x) - sin x * sin (2 * x) = 4 * cos x ^ 3 - 3 * cos x
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x
rw [← h1, cos_add x (2 * x)]
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * cos (2 * x) - sin x * sin (2 * x) = 4 * cos x ^ 3 - 3 * cos x
simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x
have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =
    2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x
rw [h2, cos_sq']
no goals
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =
    2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x
ring
no goals
x y : ℂ
⊢ x + 2 * x = 3 * x
ring
no goals
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
ring