Proposition 5

Tag 0DOC, permanent. Cite it as [Bourbaki, Tag 0DOC].

Algebra, chapter IX, § 10. Angles

Let us write $O^+$ additively. For every pair of elements $\theta, \theta'$ of $O^+$ one has

(3)

c(\theta)^2 - \delta s(\theta)^2 = 1

(4)

c(\theta + \theta') = c(\theta)c(\theta') + \delta s(\theta)s(\theta')

(5)

s(\theta + \theta') = s(\theta)c(\theta') + c(\theta)s(\theta').

Relation (3) expresses in fact that $N(c(\theta) + s(\theta)\omega) = 1$ . For (4) and (5) it is enough to calculate, in $A(\Phi)$ , the product of the rotations $c(\theta) + s(\theta)\omega$ and $c(\theta') + s(\theta')\omega$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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