/-- Finite form of Euler's sine product, with remainder term expressed as a ratio of cosine
integrals. -/
theorem sin_pi_mul_eq (z : ℂ) (n : ℕ) :
    Complex.sin (π * z) =
      ((π * z * ∏ j in Finset.range n, ((1 : ℂ) - z ^ 2 / ((j : ℂ) + 1) ^ 2)) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n)) /
        (∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n) : ℝ) := by
  rcases eq_or_ne z 0 with (rfl | hz)
  · simp
  induction' n with n hn
  · simp_rw [Nat.zero_eq, mul_zero, pow_zero, mul_one, Finset.prod_range_zero, mul_one,
      integral_one, sub_zero]
    rw [integral_cos_mul_complex (mul_ne_zero two_ne_zero hz), Complex.ofReal_zero,
      mul_zero, Complex.sin_zero, zero_div, sub_zero,
      (by push_cast; field_simp; ring : 2 * z * ↑(π / 2) = π * z)]
    field_simp [Complex.ofReal_ne_zero.mpr pi_pos.ne']
    ring
  · rw [hn, Finset.prod_range_succ]
    set A := ∏ j in Finset.range n, ((1 : ℂ) - z ^ 2 / ((j : ℂ) + 1) ^ 2)
    set B := ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n)
    set C := ∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n)
    have aux' : 2 * n.succ = 2 * n + 2 := by rw [Nat.succ_eq_add_one, mul_add, mul_one]
    have : (∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n.succ)) = (2 * (n : ℝ) + 1) / (2 * n + 2) * C := by
      rw [integral_cos_pow_eq]
      dsimp only
      rw [integral_cos_pow_eq, aux', integral_sin_pow, sin_zero, sin_pi, pow_succ,
        zero_mul, zero_mul, zero_mul, sub_zero, zero_div,
        zero_add, ← mul_assoc, ← mul_assoc, mul_comm (1 / 2 : ℝ) _, Nat.cast_mul, Nat.cast_eq_ofNat]
    rw [this]
    change
      π * z * A * B / C =
        (π * z * (A * ((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2)) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) /
          ((2 * n + 1) / (2 * n + 2) * C : ℝ)
    have :
      (π * z * (A * ((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2)) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) =
        π * z * A *
          (((1 : ℂ) - z ^ 2 / (n.succ : ℂ) ^ 2) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) := by
      nth_rw 2 [Nat.succ_eq_add_one]
      rw [Nat.cast_add_one]
      ring
    rw [this]
    suffices
      (((1 : ℂ) - z ^ 2 / (n.succ : ℂ) ^ 2) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) =
        (2 * n + 1) / (2 * n + 2) * B by
      rw [this, Complex.ofReal_mul, Complex.ofReal_div]
      have : (C : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (integral_cos_pow_pos _).ne'
      have : 2 * (n : ℂ) + 1 ≠ 0 := by
        convert (Nat.cast_add_one_ne_zero (2 * n) : (↑(2 * n) + 1 : ℂ) ≠ 0)
        simp
      have : 2 * (n : ℂ) + 2 ≠ 0 := by
        convert (Nat.cast_add_one_ne_zero (2 * n + 1) : (↑(2 * n + 1) + 1 : ℂ) ≠ 0) using 1
        push_cast; ring
      field_simp; 

Proof state:
z✝ : ℂ
n✝ : ℕ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * Nat.succ n = 2 * n + 2
this✝⁴ : ∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * Nat.succ n) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this✝³ :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * Nat.succ n) =
    ↑π * z * A *
      ((1 - z ^ 2 / ↑(Nat.succ n) ^ 2) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * Nat.succ n))
this✝² :
  (1 - z ^ 2 / ↑(Nat.succ n) ^ 2) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * Nat.succ n) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * B
this✝¹ : ↑C ≠ 0
this✝ : 2 * ↑n + 1 ≠ 0
this : 2 * ↑n + 2 ≠ 0
⊢ ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * Complex.cos ↑x ^ (2 * n)) *
      ((2 * ↑n + 1) * ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))) =
    (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ((2 * ↑n + 1) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * Complex.cos ↑x ^ (2 * n)) *
      ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))

Next tactic:
ring