Prove that in any triangle ABC, cos 2 ⁡ A + cos 2 ⁡ B + cos 2 ⁡ C ≥ 3 4

Question

Prove that in any triangle ABC,       cos    2    ⁡  A  +      cos    2    ⁡  B  +      cos    2    ⁡  C  ≥      3    4

Haggar72 · Accepted Answer

y  =      cos    2    ⁡  A  +      cos    2    ⁡  B  +      cos    2    ⁡  C  =      cos    2    ⁡  A  −      sin    2    ⁡  B  +      cos    2    ⁡  C  +  1Now       cos    2    ⁡  A  −      sin    2    ⁡  B  =  cos  ⁡  (  A  +  B  )  cos  ⁡  (  A  −  B  )  =  −  cos  ⁡  C  cos  ⁡  (  A  −  B  )    ⟹        cos    2    ⁡  C  −  cos  ⁡  C  cos  ⁡  (  A  −  B  )  +  1  −  y  =  0which is a Quadratic Equation in cosC    ⟹        cos    2    ⁡  (  A  −  B  )  −  4  (  1  −  y  )  ≥  0    ⟺    4  y  ≥  4  −      cos    2    ⁡  (  A  −  B  )  =  3  +      sin    2    ⁡  (  A  −  B  )    ⟹    4  y  ≥  3The equality occurs if   sin  ⁡  (  A  −  B  )  =  0    ⟹    A  =  B              (  1  ) and   cos  ⁡  C  =            1      2          ⟹    C  =            π      3                    (  2  )  (  1  )  ,  (  2  )    ⟹    A  =  B  =  C      cos    2    ⁡  A  +      cos    2    ⁡  B  +      cos    2    ⁡  C  =  1  +      cos    2    ⁡  C  −  cos  ⁡  C  cos  ⁡  (  A  −  B  )  =  1  +  cos  ⁡  C  [  cos  ⁡  C  −  cos  ⁡  (  A  −  B  )  ]  =  1  −  cos  ⁡  C  [  cos  ⁡  (  A  +  B  )  +  cos  ⁡  (  A  −  B  )  ]  =  1  −  2  cos  ⁡  A  cos  ⁡  B  cos  ⁡  C  &amp;lt;  1if   0  &amp;lt;  A  ,  B  ,  C  &amp;lt;            π      2

Dale Tate · Accepted Answer

Multiplying by   4      R    2   and exploiting the sine theorem we get:  4      R    2        ∑          c      y      c            cos    2    ⁡  A  =  12      R    2    −  (      a    2    +      b    2    +      c    2    )hence the first inequality is equivalent to the trivial   O      H    2    ≥  0, where   O is the circumcenter and   H is the orthocenter. On the other hand, if   A  B  C is an acute-angled triangle we have that   H lies inside   A  B  C, hence   O      H    2    &amp;lt;      R    2   and the second inequality follows.

Prove that in any triangle ABC, cos 2 </msup> ⁡ A + cos

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