Question

If a,b,c are in A.P, then prove that:
`2sinA/2.sinC/2=sinB/2`

Answer 1

Let `2sinA/2.sinC/2=sinB/2`
`rArr 2sqrt((s-b)(s-c))/(bc).sqrt((s-a)(s-b))/(ab)=sqrt((s-a)(s-c))/(ac)`
`rArr 2(s-b)/b.sqrt((s-a)(s-c))/(ac)=sqrt((s-a)(s-c))/(ac)`
`rArr 2(s-b)/(b)=1`
`rArr 2s-2b=b`
`rArr a+b+c-2b=b`
`rArr a+c=2b`
`rArr a+c=2b`
`rArr` a,b,c are in A.P.
Which is given.
Therefore, `2sinA/2.sinC/2=sinB/2`, if a,b,c are in A.P. Hence proved.