[tex] ( 1- cos(A))( 1-cos(B))(1-cos(C)) \le \ \frac{1}{8} [/tex]
SỬ DỤNG CÔNG THỨC GÓC NHÂN ĐÔI, TA ĐƯỢC:
[tex] ( Sin( \frac{A}{2})^2(sin( \frac{B}{2} )^2(sin( \frac{C}{2})^2 \le \ \frac{1}{64} [/tex]
[tex] \frac{(cos(A) + cos(B) + cos(C)-1)^2}{16} \le \ \frac{1}{64} [/tex]
[tex] cos(A) + cos(B) + cos(C) \le \ \frac{3}{2} [/tex]
[tex] - 2cos(\frac{A + B}{2}) cos(\frac{A - B}{2}) + 2sin(\frac{C}{2})^2 + \frac{1}{2} \ge \ 0 [/tex]
[tex]2( sin(\frac{C}{2})^2 - 2cos(\frac{A + B}{2}) cos(\frac{A - B}{2}) + \frac{1}{4}cos(\frac{A - B}{2})^2) - \frac{1}{2}cos(\frac{A - B}{2})^2 + \frac{1}{2} \ge \ 0 [/tex]
[tex] 2(sin(\frac{C}{2}) - cos(\frac{A - B}{2}))^2 + \frac{1}{2}( 1 - cos(\frac{A - B}{2})^2)) \ge \ 0 [/tex]
với sin(..), cos(...) <=1 . Hiển nhiên! dấu = xảy ra khi ABC cân.