The arctan function.

Inequalities, identities and Real.tan as a PartialHomeomorph between (-(π / 2), π / 2) and the whole line.

The result of arctan x + arctan y is given by arctan_add, arctan_add_eq_add_pi or arctan_add_eq_sub_pi depending on whether x * y < 1 and 0 < x. As an application of arctan_add we give four Machin-like formulas (linear combinations of arctangents equal to π / 4 = arctan 1), including John Machin's original one at four_mul_arctan_inv_5_sub_arctan_inv_239.

theorem Real.tan_add {x y : ℝ} (h : ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * Real.pi / 2) ∨ (∃ (k : ℤ), x = (2 * ↑k + 1) * Real.pi / 2) ∧ ∃ (l : ℤ), y = (2 * ↑l + 1) * Real.pi / 2) : Real.tan (x + y) = (Real.tan x + Real.tan y) / (1 - Real.tan x * Real.tan y)

theorem Real.tan_add' {x y : ℝ} (h : (∀ (k : ℤ), x ≠ (2 * ↑k + 1) * Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * Real.pi / 2) : Real.tan (x + y) = (Real.tan x + Real.tan y) / (1 - Real.tan x * Real.tan y)

theorem Real.tan_two_mul {x : ℝ} : Real.tan (2 * x) = 2 * Real.tan x / (1 - Real.tan x ^ 2)

theorem Real.tan_int_mul_pi_div_two (n : ℤ) : Real.tan (↑n * Real.pi / 2) = 0

theorem Real.continuousOn_tan : ContinuousOn Real.tan {x : ℝ | Real.cos x ≠ 0}

theorem Real.continuous_tan : Continuous fun (x : ↑{x : ℝ | Real.cos x ≠ 0}) => Real.tan ↑x

theorem Real.continuousOn_tan_Ioo : ContinuousOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))

theorem Real.surjOn_tan : Set.SurjOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) Set.univ

theorem Real.tan_surjective : Function.Surjective Real.tan

theorem Real.image_tan_Ioo : Real.tan '' Set.Ioo (-(Real.pi / 2)) (Real.pi / 2) = Set.univ

def Real.tanOrderIso : ↑(Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) ≃o ℝ

Real.tan as an OrderIso between (-(π / 2), π / 2) and ℝ.

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• One or more equations did not get rendered due to their size.

noncomputable def Real.arctan (x : ℝ) : ℝ

Inverse of the tan function, returns values in the range -π / 2 < arctan x and arctan x < π / 2

Equations

• Real.arctan x = ↑(Real.tanOrderIso.symm x)

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theorem Real.tan_arctan (x : ℝ) : Real.tan (Real.arctan x) = x

theorem Real.arctan_mem_Ioo (x : ℝ) : Real.arctan x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)

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theorem Real.range_arctan : Set.range Real.arctan = Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)

theorem Real.arctan_tan {x : ℝ} (hx₁ : -(Real.pi / 2) < x) (hx₂ : x < Real.pi / 2) : Real.arctan (Real.tan x) = x

theorem Real.cos_arctan_pos (x : ℝ) : 0 < Real.cos (Real.arctan x)

theorem Real.cos_sq_arctan (x : ℝ) : Real.cos (Real.arctan x) ^ 2 = 1 / (1 + x ^ 2)

theorem Real.sin_arctan (x : ℝ) : Real.sin (Real.arctan x) = x / √(1 + x ^ 2)

theorem Real.cos_arctan (x : ℝ) : Real.cos (Real.arctan x) = 1 / √(1 + x ^ 2)

theorem Real.arctan_lt_pi_div_two (x : ℝ) : Real.arctan x < Real.pi / 2

theorem Real.neg_pi_div_two_lt_arctan (x : ℝ) : -(Real.pi / 2) < Real.arctan x

theorem Real.arctan_eq_arcsin (x : ℝ) : Real.arctan x = Real.arcsin (x / √(1 + x ^ 2))

theorem Real.arcsin_eq_arctan {x : ℝ} (h : x ∈ Set.Ioo (-1) 1) : Real.arcsin x = Real.arctan (x / √(1 - x ^ 2))

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theorem Real.arctan_zero : Real.arctan 0 = 0

theorem Real.arctan_strictMono : StrictMono Real.arctan

theorem Real.arctan_injective : Function.Injective Real.arctan

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theorem Real.arctan_eq_zero_iff {x : ℝ} : Real.arctan x = 0 ↔ x = 0

theorem Real.tendsto_arctan_atTop : Filter.Tendsto Real.arctan Filter.atTop (nhdsWithin (Real.pi / 2) (Set.Iio (Real.pi / 2)))

theorem Real.tendsto_arctan_atBot : Filter.Tendsto Real.arctan Filter.atBot (nhdsWithin (-(Real.pi / 2)) (Set.Ioi (-(Real.pi / 2))))

theorem Real.arctan_eq_of_tan_eq {x y : ℝ} (h : Real.tan x = y) (hx : x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) : Real.arctan y = x

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theorem Real.arctan_one : Real.arctan 1 = Real.pi / 4

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theorem Real.arctan_neg (x : ℝ) : Real.arctan (-x) = -Real.arctan x

theorem Real.arctan_eq_arccos {x : ℝ} (h : 0 ≤ x) : Real.arctan x = Real.arccos (√(1 + x ^ 2))⁻¹

theorem Real.arccos_eq_arctan {x : ℝ} (h : 0 < x) : Real.arccos x = Real.arctan (√(1 - x ^ 2) / x)

theorem Real.arctan_inv_of_pos {x : ℝ} (h : 0 < x) : Real.arctan x⁻¹ = Real.pi / 2 - Real.arctan x

theorem Real.arctan_inv_of_neg {x : ℝ} (h : x < 0) : Real.arctan x⁻¹ = -(Real.pi / 2) - Real.arctan x

theorem Real.arctan_ne_mul_pi_div_two {x : ℝ} (k : ℤ) : Real.arctan x ≠ (2 * ↑k + 1) * Real.pi / 2

theorem Real.arctan_add_arctan_lt_pi_div_two {x y : ℝ} (h : x * y < 1) : Real.arctan x + Real.arctan y < Real.pi / 2

theorem Real.arctan_add {x y : ℝ} (h : x * y < 1) : Real.arctan x + Real.arctan y = Real.arctan ((x + y) / (1 - x * y))

theorem Real.arctan_add_eq_add_pi {x y : ℝ} (h : 1 < x * y) (hx : 0 < x) : Real.arctan x + Real.arctan y = Real.arctan ((x + y) / (1 - x * y)) + Real.pi

theorem Real.arctan_add_eq_sub_pi {x y : ℝ} (h : 1 < x * y) (hx : x < 0) : Real.arctan x + Real.arctan y = Real.arctan ((x + y) / (1 - x * y)) - Real.pi

theorem Real.two_mul_arctan {x : ℝ} (h₁ : -1 < x) (h₂ : x < 1) : 2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2))

theorem Real.two_mul_arctan_add_pi {x : ℝ} (h : 1 < x) : 2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2)) + Real.pi

theorem Real.two_mul_arctan_sub_pi {x : ℝ} (h : x < -1) : 2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2)) - Real.pi

theorem Real.arctan_inv_2_add_arctan_inv_3 : Real.arctan 2⁻¹ + Real.arctan 3⁻¹ = Real.pi / 4

theorem Real.two_mul_arctan_inv_2_sub_arctan_inv_7 : 2 * Real.arctan 2⁻¹ - Real.arctan 7⁻¹ = Real.pi / 4

theorem Real.two_mul_arctan_inv_3_add_arctan_inv_7 : 2 * Real.arctan 3⁻¹ + Real.arctan 7⁻¹ = Real.pi / 4

theorem Real.four_mul_arctan_inv_5_sub_arctan_inv_239 : 4 * Real.arctan 5⁻¹ - Real.arctan 239⁻¹ = Real.pi / 4

John Machin's 1706 formula, which he used to compute π to 100 decimal places.

theorem Real.continuous_arctan : Continuous Real.arctan

theorem Real.continuousAt_arctan {x : ℝ} : ContinuousAt Real.arctan x

def Real.tanPartialHomeomorph : PartialHomeomorph ℝ ℝ

Real.tan as a PartialHomeomorph between (-(π / 2), π / 2) and the whole line.

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theorem Real.coe_tanPartialHomeomorph : ↑Real.tanPartialHomeomorph = Real.tan

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theorem Real.coe_tanPartialHomeomorph_symm : ↑Real.tanPartialHomeomorph.symm = Real.arctan