Hey, this is a really neat method. I've been aware of the deMoivre method but this is definitely much easier! (especially for tangent)
However, here are some clarifications I've found on my own through trial and error:
1. When alternating your signs for all the trig functions, the numerator starts positive: + - + -... and the denominator starts negative: -+-+...
2. When deriving formulas for tangent, the first term doesn't always start on the bottom--you want to start on the side that makes the 1 go in the denominator. In more mathematical terms, start on the bottom if your expansion has an odd number of terms, and on the top if it has an even number of terms.
On that note, your expansion for tan(5θ) should be flipped so that the 1 is on the bottom.
But other than that, very nice!