......Shat.
I kind of made an assumption that's not quite valid
Here it is though, if you want to see it
Ok, so Euler's Formula states
e^(xi)=cos x + i sinx (or cis x)
Step 1: prove for x=0
e^0i= cos0 + i sin0
1=1+0i
1=1
Step 2 Assume true for x=k
e^(ki) = cis k
Step 3 Prove for x=k+1
e^(k+1)i=cis(k+1)
e^(ki+i)= //distribute i
(e^ki)(e^i)= // exponent property
(cis k)(cis 1)= // here's where i mess up. In my original proof, I did induction the standard way, except it's really hard to prove e^i=cis 1, so I just kind of assumed that it was true and used it here to sub it in as e^i=cis 1
(cos k +i sin k)(cos 1 + i sin 1)= //expand
cos k cos 1 + i sin 1 cos k + i sin k cos 1 + -1 sin k sin 1 //multiply out
cos k cos 1 -sin k sin 1 + i (sin 1 cos k + sin k cos 1) = //re order terms
cos (k+1) +i sin (k+1) //recognize cos and sin sum formulas
cis (k+1) =cis (k+1)
Subbing in pi gives e^ (pi i) = cos pi + i sin pi
= -1 + 0i
= -1
I always do this, I get an idea, I get lazy, skip a step, and then realize that it threatens the whole integrity of the proof >.<