Sums and Differences Identities Derivation
Suppose two points with coordinates of (⊖α,⦶α) and (⊖β,⦶β).
Then the distance between the two points is
√( (⊖α - ⊖β)² + (⦶α - ⦶β)² ).
It is the same as the distance between (⊖(α-β), ⦶(α-β)) and (1,0), which is
√( (⊖(α-β) - 1)² + (⦶(α-β) - 0)² ).
The former distance is
√( (⊖²α-2·⊖α·⊖β+⊖²β) + (⦶²α-2·⦶α·⦶β+⦶²β) )
which is
√( (1 - 2·⊖α·⊖β) + (1 - 2·⦶α·⦶β) ).
The latter distance is
√( (⊖²(α-β) - 2·⊖(α-β) + 1) + ⦶²(α-β) )
which is
√( 2 - 2·⊖(α-β) ).
Because the two distances are the same,
(1 - 2·⊖α·⊖β) + (1 - 2·⦶α·⦶β) = 2 - 2·⊖(α-β)
which is
⊖(α-β) = ⊖α·⊖β + ⦶α·⦶β.
_
Suppose two points with coordinates of, run of α, comma, rise of α, and run of β, comma, rise of β.
Then the distance between the two points is,
Square root of, run of α minus run of β, squared, + rise of α minus rise β, squared.
It is the same as the distance between, run of α minus β, comma, rise of α minus β, and, 1, comma, 0,
which is square root of, run of α minus β, minus 1, squared, + rise of α minus β, minus 0, squared.
The former distance is,
Square root of, run squared of α, minus, 2 times run of α times run of β, +, run squared of β, +, rise squared of α, minus, 2 times rise of α times rise of β, +, rise squared of β,
which is,
Square root of, 1, minus, 2 times run of α times run of β, +, 1, minus, 2 times rise of α times rise of β.
The latter distance is,
Square root of, run squared of, α minus β, minus, 2 times run of, α minus β, +, 1, +, rise squared of, α minus β,
which is,
Square root of, 2, minus, 2 times run of, α minus β.
Because the two distances are the same,
1, minus, 2 times run of α times run of β, +, 1, minus, 2 times rise of α times rise of β, equals, 2, minus, 2 times run of, α minus β,
which is,
Run of, α minus β, equals, run of α times run of β, +, rise of α times rise of β.
