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If-sides-a-b-c-of-ABC-are-in-H-P-prove-that-sin-2-A-2-sin-2-B-2-sin-2-C-2-are-in-H-P-

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Question Number 16041 by Tinkutara last updated on 17/Jun/17
If sides a, b, c of ΔABC are in H.P., prove that sin^2 ((A/2)), sin^2 ((B/2)), sin^2 ((C/2)) are in H.P.
Ifsidesa,b,cofΔABCareinH.P.,provethatsin2(A2),sin2(B2),sin2(C2)areinH.P.
Answered by Tinkutara last updated on 06/Jul/17
Let sin^2 ((A/2)), sin^2 ((B/2)), sin^2 ((C/2)) be in H.P. and we have to prove a, b, c are in H.P. ⇒ (1/(sin^2 (B/2))) − (1/(sin^2 (A/2))) = (1/(sin^2 (C/2))) − (1/(sin^2 (B/2))) sin^2 (C/2) (sin^2 (A/2) − sin^2 (B/2)) = sin^2 (A/2) (sin^2 (B/2) − sin^2 (C/2)) sin^2 (C/2) sin ((A + B)/2) sin ((A − B)/2) = sin^2 (A/2) sin ((B + C)/2) sin ((B − C)/2) 2 sin^2 (C/2) cos (C/2) sin ((A − B)/2) = 2 sin^2 (A/2) cos (A/2) sin ((B − C)/2) sin (C/2) sin C sin ((A − B)/2) = sin (A/2) sin A sin ((B − C)/2) ((sin A)/(sin C)) = ((2 sin (C/2) sin ((A − B)/2))/(2 sin (A/2) sin ((B − C)/2))) (a/c) = ((sin A − sin B)/(sin B − sin C)) = ((a − b)/(b − c)) ∴ (2/b) = (1/a) + (1/c)
Letsin2(A2),sin2(B2),sin2(C2)beinH.P.andwehavetoprovea,b,careinH.P.⇒1sin2B2−1sin2A2=1sin2C2−1sin2B2sin2C2(sin2A2−sin2B2)=sin2A2(sin2B2−sin2C2)sin2C2sinA+B2sinA−B2=sin2A2sinB+C2sinB−C22sin2C2cosC2sinA−B2=2sin2A2cosA2sinB−C2sinC2sinCsinA−B2=sinA2sinAsinB−C2sinAsinC=2sinC2sinA−B22sinA2sinB−C2ac=sinA−sinBsinB−sinC=a−bb−c∴2b=1a+1c

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