-4 bytes thanks to miles (use of { and } while using their formula.
,IÆẠCH;⁸ÆẠ¤Ḣ+PƊ½ÆṢ×⁽µṣ;I}ÆẠP+ÆSP${ÆA×⁽£ġ
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How?
,IÆẠCH;⁸ÆẠ¤Ḣ+PƊ½ÆṢ×⁽µṣ - Link: [ϕ1, ϕ2]; [λ1, λ2]
, - pair = [[ϕ1, ϕ2], [λ1, λ2]]
I - incremental differences = [ϕ2-ϕ1, λ2-λ1]
ÆẠ - cosine = [cos(ϕ2-ϕ1), cos(λ2-λ1)]
C - complement = [1-cos(ϕ2-ϕ1), 1-cos(λ2-λ1)]
H - halve = [sin²((ϕ2-ϕ1)/2), sin²((λ2-λ1)/2)]
¤ - nilad followed by links as a nilad:
⁸ - chain's left argument = [ϕ1, ϕ2]
ÆẠ - cosine = [cos(ϕ1), cos(ϕ2)]
; - concatenate = [sin²((ϕ2-ϕ1)/2), sin²((λ2-λ1)/2), cos(ϕ1), cos(ϕ2)]
Ɗ - last three links as a monad:
Ḣ - remove head and yield = sin²((ϕ2-ϕ1)/2)
P - product = sin²((λ2-λ1)/2) × cos(ϕ1) × cos(ϕ2)
+ - add = sin²((ϕ2-ϕ1)/2) + sin²((λ2-λ1)/2) × cos(ϕ1) × cos(ϕ2)
½ - square-root = √(sin²((ϕ2-ϕ1)/2) + sin²((λ2-λ1)/2) × cos(ϕ1) × cos(ϕ2))
ÆṢ - arcsine = asin(√(sin²((ϕ2-ϕ1)/2) + sin²((λ2-λ1)/2) × cos(ϕ1) × cos(ϕ2)))
⁽µṣ - literal 3474
× - multiply = 3474×asin(√(sin²((ϕ2-ϕ1)/2) + sin²((λ2-λ1)/2) × cos(ϕ1) × cos(ϕ2)))
GolfingMine... (miles') alternate form,also saved a byte here by using a r × acos(sin(ϕ1) × sin(ϕ2) + cos(ϕ1) × cos(ϕ2) × cos(λ2 - λ1)){, I got this 23 byter:)
ÆSżÆẠƊP;⁹IÆẠ¤Ḣ+PƊÆA×⁽£ġ,IÆẠCH;ÆẠ{Ḣ+PƊ½ÆṢ×⁽µṣ
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