<Tri-2025-2283>, hint

Let M be the midpoint of OH, O' be the circumcenter of triangle HOA'

Height from BC  to midline m parallel to BC ; 1/2 (AA')=RsinC sinB 

Height  from BC to H: 2RsinC cosB cotC=2R cosBcosC 

Height from BC to O=R cosA=-R(cos(B+C)

Height  from BC to O' ( circumcenter of HOA')=HA'/2=RcosB cosC

Since OM=MH and O'M=MP(P; point on midleline m)

1/2(RcosA+2R cosB cosC)=1/2(R cosB cosC+RsinC sin B) --->

-cos(B+C)+cosBcosC=sinBSinC <---> -cosBcosC+sinBsinC=sinBsinC-cosBcosC