1.

In a triangle ABC `sin (A/2) sin (B/2) sin (C/2)

Answer» We know, arithmatic mean is always greater than or equal to geometric mean.
`:. (sin^2(A/2)sin^2(B/2)sin^2(C/2))^(1/3) le 1/3(sin^2(A/2)+sin^2(B/2)+sin^2(C/2))->(1)`
Also, we know, `sin^2(A/2)+sin^2(B/2)+sin^2(C/2) ge 3/4`
`:. (sin^2(A/2)sin^2(B/2)sin^2(C/2))^(1/3) le 1/3(3/4)`
`=>(sin^2(A/2)sin^2(B/2)sin^2(C/2))^(1/3) le 1/4`
`=>(sin^2(A/2)sin^2(B/2)sin^2(C/2)) le (1/4)^3`
`=>sin(A/2)sin(B/2)sin(C/2) le (1/64)^(1/2)`
`=>sin(A/2)sin(B/2)sin(C/2) le 1/8`


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