If A and B be the greatest and least angles of a triangle whose sides
are in A.P, then
1)4(1-sins)(1-sinB)=cosA+cosB
2)4(1-cosA)(1-cosB)=cosA+cosB
3)4(1+cosA)(1+cosB)=cosA+cosB
4)None of these

Asked by Sunil Soni | 7th Dec, 2015, 11:46: PM

Expert Answer:

begin mathsize 11px style Given space that space the space sides space of space straight a space triangle space are space in space straight A. straight P. Since space straight A space and space straight B space are space greatest space and space least space angles space respectively comma straight a space and space straight b space are space longest space and space shortest space sides space respectively. Hence comma space straight b comma space straight c space and space straight a space are space in space straight A. straight P. rightwards double arrow straight a plus straight b equals 2 straight c... left parenthesis 1 right parenthesis From space sine space rule comma space straight a over sinA equals straight b over sinB equals straight c over sinC equals straight K rightwards double arrow straight a equals KsinA comma space straight b equals KsinB comma space straight c equals KsinC Substituting space the space above space in space left parenthesis 1 right parenthesis comma space we space get sinA plus sinB equals 2 sinC rightwards double arrow 2 sin fraction numerator straight A plus straight B over denominator 2 end fraction cos fraction numerator straight A minus straight B over denominator 2 end fraction equals 2 cross times 2 sin straight C over 2 cos straight C over 2 rightwards double arrow 2 cos straight C over 2 cos fraction numerator straight A minus straight B over denominator 2 end fraction equals 4 sin straight C over 2 cos straight C over 2 rightwards double arrow cos fraction numerator straight A minus straight B over denominator 2 end fraction equals 2 sin straight C over 2.... left parenthesis 2 right parenthesis  RHS space of space the space options space is space cosA plus cosB equals 2 cos fraction numerator straight A plus straight B over denominator 2 end fraction cos fraction numerator straight A minus straight B over denominator 2 end fraction equals 2 cos fraction numerator straight A plus straight B over denominator 2 end fraction cross times 2 sin straight C over 2 equals 4 sin squared straight C over 2.... left parenthesis 3 right parenthesis  LHS space of space option space left parenthesis 2 right parenthesis space is space 4 open parentheses 1 minus cosA close parentheses open parentheses 1 minus cosB close parentheses equals 4 cross times 2 sin squared straight A over 2 cross times 2 sin squared straight B over 2 equals 4 open parentheses 2 sin straight A over 2 sin straight B over 2 close parentheses squared equals 4 open parentheses cos fraction numerator straight A minus straight B over denominator 2 end fraction minus cos fraction numerator straight A plus straight B over denominator 2 end fraction close parentheses squared equals 4 open parentheses 2 sin straight C over 2 minus sin straight C over 2 close parentheses squared equals 4 sin squared straight C over 2... left parenthesis 4 right parenthesis From space equation space left parenthesis 3 right parenthesis space and space left parenthesis 4 right parenthesis 4 open parentheses 1 minus cosA close parentheses open parentheses 1 minus cosB close parentheses equals cosA plus cosB Hence comma space 2 nd space option space is space correct.  end style

Answered by satyajit samal | 8th Dec, 2015, 07:11: PM