tan² θ + 1 = sec² θ but only if cos² θ ≠ 0??

TrigonometricAnalysis.lean

@@ -0,0 +1,89 @@
+-- actually it's analytical trigonometry
+import Mathlib.Data.Real.Basic
+import Mathlib.Tactic.LibrarySearch
+
+axiom sin : Real → Real 
+axiom csc : Real → Real 
+
+axiom sin_eq_reciprocal_of_csc : ∀x, sin x = 1 / (csc x)
+axiom csc_eq_reciprocal_of_sin : ∀x, csc x = 1 / (sin x)
+
+axiom cos : Real → Real 
+axiom sec : Real → Real 
+
+axiom cos_eq_reciprocal_of_sec : ∀x, cos x = (sec x)⁻¹ 
+axiom sec_eq_reciprocal_of_cos : ∀x, sec x = (cos x)⁻¹
+
+@[simp]
+noncomputable def tan (x : Real) : Real := sin x / cos x
+theorem tan_def : ∀x, tan x = sin x / cos x := by simp
+@[simp]
+noncomputable def cot (x : Real) : Real := cos x / sin x
+
+theorem tan_eq_sin_div_cos : ∀x, tan x = sin x / cos x := by simp
+theorem cot_eq_cos_div_sin : ∀x, cot x = cos x / sin x := by simp
+
+theorem tan_eq_reciprocal_of_cot : ∀x, tan x = 1 / (cot x) := by simp
+theorem cot_eq_reciprocal_of_tan : ∀x, cot x = 1 / (tan x) := by simp
+
+noncomputable def sin2 (x : Real) : Real := sin x * sin x
+noncomputable def cos2 (x : Real) : Real := cos x * cos x
+noncomputable def tan2 (x : Real) : Real := tan x * tan x
+theorem tan2_def : ∀x, tan2 x = sin2 x / cos2 x := by 
+  intro x
+  simp [tan2, sin2, cos2]
+  exact div_mul_div_comm (sin x) (cos x) (sin x) (cos x)
+
+noncomputable def csc2 (x : Real) : Real := csc x * csc x
+noncomputable def sec2 (x : Real) : Real := sec x * sec x
+theorem sec2_eq_reciprocal_of_cos2 : ∀x, sec2 x = (cos2 x)⁻¹ := by 
+  intro x
+  rw [sec2, cos2]
+  simp
+  rw [sec_eq_reciprocal_of_cos]
+
+noncomputable def cot2 (x : Real) : Real := cot x * cot x
+
+noncomputable def sin3 (x : Real) : Real := sin2 x * sin x
+noncomputable def cos3 (x : Real) : Real := cos2 x * cos x
+noncomputable def tan3 (x : Real) : Real := tan2 x * tan x
+
+noncomputable def csc3 (x : Real) : Real := csc2 x * csc x
+noncomputable def sec3 (x : Real) : Real := sec2 x * sec x
+noncomputable def cot3 (x : Real) : Real := cot2 x * cot x
+
+@[simp]
+axiom pyth_id : ∀x, sin2 x + cos2 x = 1
+
+theorem sin_squared_eq {x : Real} : sin2 x = 1 - cos2 x := by 
+  rw [← add_left_inj (cos2 x)]
+  rw [sub_add_cancel]
+  exact pyth_id x
+
+theorem cos_squared_eq {x : Real} : cos2 x = 1 - sin2 x := by 
+  rw [← add_left_inj (sin2 x)]
+  rw [sub_add_cancel]
+  rw [add_comm]
+  exact pyth_id x
+
+theorem bleh {a b c : Real} (d : Real) : a + b = c → a / d + b / d = c / d := by
+  intro h
+  rw [← add_div, h]
+
+def tan_pyth_id : ∀x, cos2 x ≠ 0 → tan2 x + 1 = sec2 x := by 
+  intro x
+  intro hnz
+  have h := pyth_id x
+  have h : sin2 x / cos2 x + cos2 x / cos2 x = 1 / cos2 x :=
+    bleh (cos2 x) h
+
+  rw [← tan2_def] at h
+  simp at h
+  rw [← sec2_eq_reciprocal_of_cos2, div_self] at h
+  exact h
+  exact hnz
+
+example (x : Real) : 2 * (1 / (csc x)) = 2 * (sin x) := by
+  rw [sin_eq_reciprocal_of_csc]
+
+--todo evens and odds and the sin/cos (a ± b) ones

lake-manifest.json

@@ -0,0 +1,27 @@
+{"version": 4,
+ "packagesDir": "lake-packages",
+ "packages":
+ [{"git":
+   {"url": "https://github.com/leanprover-community/mathlib4.git",
+    "subDir?": null,
+    "rev": "076ab7d4b2f10fcbb8d8030c608caa254a3f77af",
+    "name": "mathlib",
+    "inputRev?": null}},
+  {"git":
+   {"url": "https://github.com/gebner/quote4",
+    "subDir?": null,
+    "rev": "7ac99aa3fac487bec1d5860e751b99c7418298cf",
+    "name": "Qq",
+    "inputRev?": "master"}},
+  {"git":
+   {"url": "https://github.com/JLimperg/aesop",
+    "subDir?": null,
+    "rev": "ba61f7fec6174d8c7d2796457da5a8d0b0da44c6",
+    "name": "aesop",
+    "inputRev?": "master"}},
+  {"git":
+   {"url": "https://github.com/leanprover/std4",
+    "subDir?": null,
+    "rev": "de7e2a79905a3f87cad1ad5bf57045206f9738c7",
+    "name": "std",
+    "inputRev?": "main"}}]}

lakefile.lean

@@ -0,0 +1,14 @@
+import Lake
+open Lake DSL
+
+package «trigonometric-analysis» {
+  -- add any package configuration options here
+}
+
+require mathlib from git
+  "https://github.com/leanprover-community/mathlib4.git"
+
+@[default_target]
+lean_lib «TrigonometricAnalysis» {
+  -- add any library configuration options here
+}

lean-toolchain

@@ -0,0 +1 @@
+leanprover/lean4:nightly-2023-02-10