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

Answer

7

Explanation

`2 sin ""(A)/(2) sin ""(B)/(2) sin ""(C )/(2)= (1)/(16)`
`rArr ( cos ""(A-B)/(2) - cos ""(A+B)/(2)) sin ""(C )/(2) = (1)/(16)`
or `sin ""(C )/(2) ( (1)/(2) - sin ""(C )/(2))= (1)/(16)`
or `sin ^(2)""(C )/(2) -(1)/(2) sin""(C )/(2) + (1)/(16) = 0 `
or `((1)/(4) - sin ""(C ) /(2))^(2) =0`
or `sin""(C )/(2) = (1)/(4)`
or `cos C =1- 2 sin ^(2) ""(C )/(2) = 1 - (1)/(8) = (7)/(8)`