In a triangle `A B C ,` if `A-B=120^0a n dsin(A/2)sin(B/2)sin(C/2)=1/(32),` then the value of `8cosC` is_______

Explanation

`sin(A/2)sin(B/2)sin(C/2) = 1/32`

`=>(2sin(A/2)sin(B/2))sin(C/2) = 2/32`

`=>[cos((A-B)/2) - cos((A+B)/2)]sin(C/2) = 1/16`

`=>[cos(120^@/2) - cos((A+B)/2)]sin(C/2) = 1/16`

`=>[1/2 - cos((pi-C)/2)]sin(C/2) = 1/16`

`=>[1/2 - sin(C/2)]sin(C/2) = 1/16`

`=>1/2sin(C/2) - sin^2(C/2) = 1/16`

`=>sin^2(C/2) - 1/2sin(C/2) +(1/4)^2 = 0`

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