Tangent of a Difference: Trig Identity Puzzle + Proof

GeoGebra Commands & Parameters

1. Rotate(E, (a * π / 180), A) → B'
2. Angle(E, A, B') → α
3. Ray(A, B') → g
4. Rotate(E, (b * π / 180), A) → B'_1
5. Angle(E, A, B'_1) → β
6. Ray(A, B'_1) → h
7. Intersect(f, h) → D
8. Intersect(g, f) → C
9. Line(D, A) → i
10. Line(C, A) → j
11. Circle(A, 1) → c
12. Intersect(c, j, 2) → F_1
13. Ray(A, F_1) → l
14. Ray(A, (-1, 0)) → m
15. Angle(G, A, F_1) → γ
16. Rotate(E, α, D) → B'_2
17. Angle(E, D, B'_2) → δ
18. Ray(D, B'_2) → k
19. Intersect(n, k) → F
20. Rotate(D, π, D) → C'
21. Rotate(B'_2, π, D) → H
22. Ray(C', H) → k'
23. Intersect(g, k') → B
24. Segment(B, D) → p
25. Segment(D, E) → q
26. Segment(E, F) → r
27. Min({p, q, r}) → d
28. Circle(A, (0.6 * d)) → e
29. Circle(D, (0.6 * d)) → s
30. Intersect(e, l) → J,
31. Intersect(n, m) → K
32. Intersect(e, n) → L, M
33. Intersect(e, h) → N,
34. Intersect(e, g) → O,
35. CircleSector(A, L, J) → t
36. CircleSector(A, M, N) → c_1
37. CircleSector(A, N, O) → d_1
38. Distance(A, E) → distanceAB
39. Angle(D, B, C) → ζ
40. Polygon(B, D, C) → t1, d_2, i_1, c_2
41. Polygon(D, E, F) → t2, e_1, c_3, b_1
42. Polygon(A, E, D, B) → q1, a_1, b_2, c_4, i_2
43. Intersect(f_1, n) → P
44. Intersect(f_1, l) → Q
45. Polygon(A, P, Q) → t3, q_1, a_2, p_1
46. Segment(P, Q) → g_1
47. Segment(A, P) → h_1
48. Segment(A, Q) → j_1
49. Segment(B, F) → k_1
50. Segment(A, C) → l_1
51. Segment(E, F) → n_1
52. Segment(A, E) → r_1
53. Rotate(t, If[0 < o ≤ 1, (o * π / 2), π / 2], A) → t'
54. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], D) → A'
55. Vector(A, A') → u
56. Translate(t', u) → t''
57. Angle(F, E, C) → ε
58. Segment(A, D) → s_1
59. Dilate(E, If[2 < o ≤ 3, 3 - o, 0], F) → B'_3
60. Vector(E, B'_3) → v
61. Translate(C, v) → D'
62. Translate(E, v) → B'_4
63. Segment(D', B'_4) → t_1
64. Angle(R, F, D') → η
65. Segment(C, D) → m_1
66. Segment(D, E) → f_2
67. Ray(F, R) → g_2