After thinking I've found what you're talking about I completely agree variables come out of nowhere and aren't defined, the bigger version of his worksheet is very hard to follow and I'm sure there are a couple more variables such as X instead of x which appear from nowhere too.

The only bit I can work out is that when theta is small you can say

cos(theta) = 1 - (theta^2)/2

and if you replace theta with n theta (assuming it's still always small enough)

cos(n * theta) = 1 - (n * theta^2)/2


WAIT ONE MINUTE!!!! (and this isn't the reason but it makes a link)

If you draw a right angled triangle with cos(y) = (1+y)/(1-y) you can say the hyp = 1+y and the adj = 1-y

If you then try and work out what the opposite side is you can find out that it is 2root(y)