Question

In quadrilateral `A B C D ,` if `sin((A+B)/2)cos((A-B)/2)+"sin"((C+D)/2)cos((C-D)/2)=2` then find the value of `sinA/2sinB/2sinC/2sinD/2dot`

Answer 1

`1/2[2sin((A+B)/2)cos((A-B)/2)+2sin((C+D)/2)cos((C-D)/2)]=2`
`2sinCcosD=sin(C+D)+sin(C-D)`
`sinA+sinB+sinC+sinD=4`
`sinA-11=sinB=sinC=sinD`
`A=pi/2=B=C=D`
`sin(A/2)sin(B/2)sin(C/2)sin(D/2)`
`=sin^4pi/4=(1/sqrt2)^4=1/4`.