Geometric proof for deriative of sin: How do we know both angles are congruent?

Question

In the diagram below (from https://mrchasemath.com/2017/04/03/derivatives-of-trigonometric-functions/ )

How do we know that both angles marked $x$ (with the double lines) are congruent?

Answer 1

Note that the red arc and the green line are not both straight lines, so a-priori, it doesn’t make sense to talk about the angle between the two, since the notion of angle is defined between a pair of intersecting lines. So, what we can only talk about is the angle formed by the green line and the tangent to the red arc (refer to the picture below):

What we can talk about is the angle $SPR$, which is the angle formed by the tangent to the circle at $P$ and the line segment $PR$. Now, notice that \begin{align} \angle RPS =\pi-\angle SPO - \angle OPA =\pi-\frac{\pi}{2}-\left(\frac{\pi}{2}-\angle POA\right) =\angle POA =x, \end{align} where in the second equal sign, I used $\angle SPO=\frac{\pi}{2}$ because for a circle, the radius $OP$ is perpendicular to its tangent $SP$.

Having said this, let me warn you that this picture, as nice as it is, and the subsequent calculations you present, do not constitute a proof that $\sin’(x)=\cos x$. Because there are too many gaps to be filled in. The “proof” does not carefully explain why in the limit the red arc can be approximated by the purple tangent line, i.e the second step in your chain of equalities is unjustified.