PROOF 149

Prove.

Two reflections make a rotation

On the Euclidean plane.

What you may use

A1expansions of cos(s±t), sin(s±t)
addition theorems for trigonometric functions (Proof 35) — cos s cos t ∓ sin s sin t and sin s cos t ± cos s sin t
A2sin(−t) = −sin t
sine is odd (Proof 35)
A3P = (r cos α, r sin α)
polar coordinates — distance r from the origin, angle α with the positive x-axis
A4reflections preserve distance and angle
Proof 148
A5origin and axes are chosen freely
choice of coordinate system — any point may serve as origin and any direction as x-axis
A6length of a vertical segment = |Δy|
length of a segment parallel to an axis (Proof 21)
A7two lines meet in at most one point
Euclid's first postulate — since only one line passes through two points
A8a(b+c) = ab + ac
distributive law
A9(a+b)+c = a+(b+c)
associativity of addition
A10a + b = b + a, ab = ba
commutativity of addition and multiplication
A11a−b=a+(−b), −(−a)=a
convention of subtraction and signs
A12a(−b) = −(ab)
sign rule
A13−(a+b) = (−a) + (−b)
distribution of the sign
A14cos 90° = 0, sin 90° = 1
trigonometric values at 90°
A15cos 0 = 1, sin 0 = 0
trigonometric values at 0
A162a = a + a
doubling
A17t ≠ 0 ⟹ 2t ≠ 0
twice a nonzero number is nonzero
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

Let ℓ_θ be the line through the origin making angle θ with the positive x-axis. Define two transformations in coordinates. σ_θ(x, y) = (x cos 2θ + y sin 2θ, x sin 2θ − y cos 2θ), R_φ(x, y) = (x cos φ − y sin φ, x sin φ + y cos φ)

Proof.

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