/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
    have H3 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    have H4 :
      ⟪y, y⟫ * (⟪y, y⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪x, x⟫)) - (⟪y, y⟫ - ⟪x, y⟫) * (⟪y, y⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    rw [right_distrib, right_distrib, right_distrib, right_distrib, H1, sin_angle_mul_norm_mul_norm,
      norm_sub_rev x y, H2, sin_angle_mul_norm_mul_norm, norm_sub_rev y x,
      mul_assoc (Real.sin (angle x y)), sin_angle_mul_norm_mul_norm, inner_sub_left, inner_sub_left,
      inner_sub_right, inner_sub_right, inner_sub_right, inner_sub_right, real_inner_comm x y, H3,
      H4, real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm,
      real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
    field_simp [hxn, hyn, hxyn]
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
H3 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
H4 :
  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ (Real.sqrt
                  (‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) * (2 * 2) -
                    (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖)) *
                (‖y‖ * ‖y‖ * 2 - (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖)) *
              ‖y‖ *
            ‖x - y‖ *
          (2 * (‖x‖ * ‖x - y‖) * 2) +
        (‖x‖ * ‖x‖ * 2 - (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖)) *
                Real.sqrt
                  (‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) * (2 * 2) -
                    (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖)) *
              ‖x‖ *
            ‖x - y‖ *
          (2 * (2 * (‖y‖ * ‖x - y‖)))) *
      2 =
    Real.sqrt
            (‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) * (2 * 2) -
              (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖)) *
          ‖x - y‖ *
        ‖x - y‖ *
      (2 * (2 * (‖y‖ * ‖x - y‖)) * (2 * (‖x‖ * ‖x - y‖) * 2))

Next tactic:
ring