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Question

In ABC, if sin2A2,sin2C2 be in H. P. then a, b, c will be in 
  1. A. P.

  2. G. P.

  3. H. P.
  4. None of these


Solution

The correct option is C H. P.
1sin2C21sin2B2=1sin2C2areinA.P.
1sin2C21sin2B2=1sin2B21sin2A2
ab(sa)(sb)ac(sa)(sc)
=ac(sa)(sc)bc(sb)(sc)
(asa)(b(sc)c(sb)(sb)(sc))=(csc)(a(sb)b(sa)(sa)(sb))
abs - abc - acs + abc = acs - abc - bcs + abc
ab - ac = ac - bc ab + bc = 2ac
or, 1c+1a=2b,i.e.,a,b,careinH.P.
Note : Students should remember this question as a fact.

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