Substitute - That part where the notes don't exist

I had a realization earlier today; I don't look at my notes for Substitute anymore.

As you all can imagine, when you start out on a story like this, you make notes. You make quite a few notes.

I mean, after all, complicated time and space doesn't just happen out of the blue. It does require some thought, and it does require some coordination. So I had a bunch of written notes that helps me piece all of this together.

And I'm well past the point where I need to reference any of my notes, but I only now realized that I'm past that point. After you've run it through your head enough times, you don't even need to look at the notes anymore.

And I'm so near the end of the story anyhow that I wouldn't need to anyways. But, regardless, that happened.

And now, because I can, I will prove cos(θ)² + sin(θ)² = 1 (which I proved in a previous blog but will explore an alternative method)

cos(x) = (eᶦˣ + e-ᶦˣ)/ 2
sin(x) = (eᶦˣ - e-ᶦˣ)/ 2i

sin(x)² + cos(x)²
= ((eᶦˣ - e-ᶦˣ)/ 2i)² + ((eᶦˣ + e-ᶦˣ)/ 2)²
= ((eᶦˣ - e-ᶦˣ)(eᶦˣ - e-ᶦˣ)/ -4) + ((eᶦˣ + e-ᶦˣ)(eᶦˣ + e-ᶦˣ)/ 4)
= ((eᶦˣeᶦˣ - eᶦˣe-ᶦˣ) - (eᶦˣe-ᶦˣ - e-ᶦˣe-ᶦˣ)) / -4) + ((eᶦˣeᶦˣ + eᶦˣe-ᶦˣ) + (e-ᶦˣeᶦˣ + e-ᶦˣe-ᶦˣ)) / 4)
= ((eᶦ²ˣ - 1) - (1 - e-ᶦ²ˣ)) / -4) + ((eᶦ²ˣ + 1) + (1 + e-ᶦ²ˣ)) / 4)
= (eᶦ²ˣ - 2 + e-ᶦ²ˣ)) / -4) + (eᶦ²ˣ + 2 + e-ᶦ²ˣ) / 4)
= (-eᶦ²ˣ + 2 - e-ᶦ²ˣ + eᶦ²ˣ + 2 + e-ᶦ²ˣ) / 4
= 4 / 4
= 1

Thus, sin(x)² + cos(x)² = 1
Q.E.D.