PROOF 35

Prove.

The angle addition theorems

On the unit circle.

What you may use

A1d² = (Δx)² + (Δy)²
Distance between two points (Proof 21)
A2rotation preserves distance
Rotation about the origin (cited here, not proved)
A3rotating P(θ) by φ gives P(θ+φ)
Rotation adds angles
A4P(θ)=(cos θ, sin θ)
Point on the unit circle (definition of Proof 33)
A5sin²θ + cos²θ = 1
Proof 33
A6cos 0 = 1, sin 0 = 0
Trigonometric functions of 0
A7cos(−θ) = cos θ
Cosine is even (cited here, not proved)
A8sin(−θ) = −sin θ
Sine is odd (cited here, not proved)
A9cos(90°−θ) = sin θ
Complementary angle (cited here, not proved)
A10sin(90°−θ) = cos θ
Complementary angle (cited here, not proved)
A11(u−v)² = u²−2uv+v²
Product formula (Proof 23)
A12a(b + c) = ab + ac
Distributive law
A13a + b = b + a, ab = ba
Commutativity of addition and multiplication
A14(a+b)+c = a+(b+c)
Associativity of addition
A15a × 1 = a, a × 0 = 0
Multiplication by 1 and 0
A16a + 0 = a
Additive identity
A17a(−b) = −(ab)
Sign rule 1
A18−(−a) = a, −a−b = −(a+b)
Sign rule 2
A19a + (−a) = 0
Additive inverse
A20a + b = a + c ⟹ b = c
Cancellation law for addition
A21ca = cb, c ≠ 0 ⟹ a = b
Cancellation law for multiplication
A22−2 ≠ 0
−2 is nonzero
Ea = b ⟹ same operation on both sides
Properties of equality

Nothing outside this list may be cited.

Definition D1

a − b means a + (−b), a² means aa, and 2 means 1 + 1.

Proof.

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