√(r²+d²-2rdcos(x))=
r(1+(d²-2rdcos(x))/r²)^½≃(using binomial expansion to first degree):
r(1+(d²-2rdcos(x))/(2r²))=
r+(d²-2rdcos(x))/(2r)=
r+½d²/r-dcos(x).
The accuracy of the approximation improves the smaller d is in comparison to r.
The expression reduces further to r-dcos(x).
The original radical appears to be an application of the Cosine Rule to find the length of the third side of a triangle when the lengths of the other two sides and the included angle are known.