PROOF 33

Prove.

sin²θ + cos²θ = 1

On the unit circle.

What you may use

A1In a right triangle, a² + b² = c²
Pythagorean theorem (cited here, not proved)
A2trigonometric ratios depend on the angle alone
Proof 32
A3a right triangle with acute angle θ exists
For every θ with 0° < θ < 90°
A4unit circle = {P : OP = 1}
The circle of radius 1 centered at the origin O
A5for each angle θ, P(θ) is unique
Existence and uniqueness of the point on the unit circle
A60<θ<90° ⟹ coordinates of P(θ) > 0
The point of an acute angle lies in the first quadrant
A7horizontal: |x|, vertical: |y|
Length of a segment parallel to an axis (from the origin)
A8horizontal segment ⊥ vertical segment
The coordinate axes are perpendicular
A9|t|² = t², t≥0 ⟹ |t| = t
Absolute value
A10a ÷ 1 = a, 1² = 1
Arithmetic of 1
A11a + 0 = a, 0² = 0
Arithmetic of 0
A12a + b = b + a
Commutativity of addition

Nothing outside this list may be cited.

Definition D1

For an acute angle θ, in a right triangle having θ, sin θ is the side opposite θ divided by the hypotenuse, and cos θ is the side adjacent to θ divided by the hypotenuse.

Proof.