You asked a natural question:

What if we defined cos and pi according to this formula? How could we then prove that the cosine function gives us the ratio of adj/hyp for a right angled triangle, and pi is the ratio of a circle's circumference to its diameter?

One way is to then define the sine function as $\;\sin(\theta) := \cos(\theta-\pi/2).\;$ Then verify that $\;\sin^2(\theta)+\cos^2(\theta)^2=1,\;$ and assuming the Pythagorean theorem, that the function $\;w: \theta \mapsto (\cos(\theta),\sin(\theta))\;$ parameterizes a unit circle. Each point in the first quadrant of the-circle uniquely determines a right triangle with the angle $\;\theta\;$ at the origin $\;(0,0)\;$ and the right angle at $\;(\cos(\theta),0).\;$ The hypotenuse has length $1$ and the adjacent side has length $\;\cos(\theta).\;$

For the circle circumference, the wrapping function $\;w\;$ maps to points on the circle with constant angular speed. This can be proved from $\;\cos'(\theta)=-\sin(\theta)\;$ and $\;\sin'(\theta)=\cos(\theta)\;$ using the original differential equation. Now this implies that the angular speed is one unit of arc for one unit of $\theta.$ As $\;theta\;$ goes from $\;0\;$ to  $\;2\pi\;$ to travel along the complete circumference, the circumference must be equal to $\;2\pi.$

You asked a natural question:

What if we defined cos and pi according to this formula? How could we then prove that the cosine function gives us the ratio of adj/hyp for a right angled triangle, and pi is the ratio of a circle's circumference to its diameter?

One way is to then define the sine function as $\;\sin(\theta) := \cos(\theta-\pi/2).\;$ Then verify that $\;\sin^2(\theta)+\cos^2(\theta)^2=1,\;$ and assuming the Pythagorean theorem, that the function $\;w: \theta \mapsto (\cos(\theta),\sin(\theta))\;$ parameterizes a unit circle. Each point in the first quadrant of the-circle uniquely determines a right triangle with the angle $\;\theta\;$ at the origin $\;(0,0)\;$ and the right angle at $\;(\cos(\theta),0).\;$ The hypotenuse has length $1$ and the adjacent side has length $\;\cos(\theta).\;$

For the circle circumference, the wrapping function $\;w\;$ maps to points on the circle with constant angular speed. This can be done using the original differential equation to prove that $\;\cos'(\theta)=-\sin(\theta)\;$ and $\;\sin'(\theta)=\cos(\theta)\;$. Now this implies that the angular speed is one unit of arc for one unit of $\theta.$ As $\;\theta\;$ goes through one period from $\;0\;$ to  $\;2\pi\;$ to travel along the complete circumference, this implies that the circumference must be equal to $\;2\pi.$ There are a few details that could be proved better, but this is essentially what is required.

You asked a natural question:

What if we defined cos and pi according to this formula? How could we then prove that the cosine function gives us the ratio of adj/hyp for a right angled triangle, and pi is the ratio of a circle's circumference to its diameter?

One way is to then define the sine function as $\;\sin(\theta) := \cos(\theta-\pi).\;$ Then verify that $\;\sin^2(\theta)+\cos^2(\theta)^2=1,\;$ and assuming the Pythagorean theorem, that the function $\;\theta \mapsto (\cos(\theta),\sin(\theta))\;$ parameterizes a unit circle. Each point on the upper semi-circle uniquely determines a right triangle with the angle $\;\theta\;$ at the origin $\;(0,0)\;$ and the right angle at $\;(\cos(\theta),0).\;$ The hypotenuse has length $1$ and the adjacent side has length $\;\cos(\theta).\;$ I suppose there are a few details that need proving, but essentially this is it.

You asked a natural question:

What if we defined cos and pi according to this formula? How could we then prove that the cosine function gives us the ratio of adj/hyp for a right angled triangle, and pi is the ratio of a circle's circumference to its diameter?

One way is to then define the sine function as $\;\sin(\theta) := \cos(\theta-\pi/2).\;$ Then verify that $\;\sin^2(\theta)+\cos^2(\theta)^2=1,\;$ and assuming the Pythagorean theorem, that the function $\;w: \theta \mapsto (\cos(\theta),\sin(\theta))\;$ parameterizes a unit circle. Each point in the first quadrant of the-circle uniquely determines a right triangle with the angle $\;\theta\;$ at the origin $\;(0,0)\;$ and the right angle at $\;(\cos(\theta),0).\;$ The hypotenuse has length $1$ and the adjacent side has length $\;\cos(\theta).\;$

For the circle circumference, the wrapping function $\;w\;$ maps to points on the circle with constant angular speed. This can be proved from $\;\cos'(\theta)=-\sin(\theta)\;$ and $\;\sin'(\theta)=\cos(\theta)\;$ using the original differential equation. Now this implies that the angular speed is one unit of arc for one unit of $\theta.$ As $\;theta\;$ goes from $\;0\;$ to $\;2\pi\;$ to travel along the complete circumference, the circumference must be equal to $\;2\pi.$