Cho tam giác ABC. Chứng minh sinA/2 . sinB/2 . sinC/2 ≤ 1/8

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1. `=>` Tặng bạn 

   Đặt   `P = sinA/2.sinB/2.sinC/2`.

   `<=> 2P = ((2sinA)/2*(2.sinB)/2).(2sinC)/2 `

   `<=> 2P = [cos(A/2-B/2) – cos(A/2+B/2)].sin(C/2) `

   `<=> 2P = [cos(A/2-B/2) – sin(C/2)].sin(C/2)` `<=> 2P= sin(C/2).cos(A/2-B/2) – sin²(C/2)` 

   `<=> 8P = 4sin(C/2).cos(A/2-B/2) – 4sin²(C/2)` 

   `<=> 1-8P = 4sin²(C/2) – 4sin(C/2).cos(A/2-B/2) + cos²(A/2-B/2) + 1 – cos²(A/2-B/2) `

   `<=> 1-8P = [2sin(C/2) – cos(A/2-B/2)]² + sin²(A/2-B/2) ≥ 0 (*) `

   `=> P ≤ 1/8`

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