If a , b , c ∈ ( 0 , π 2 ) ,, Then prove that sin ⁡ ( a + b + c ) sin ⁡ a + sin ⁡ b + sin ⁡ c &lt; 1

Question

If   a  ,  b  ,  c  ∈      (    0    ,          π      2        )      ,, Then prove that             sin      ⁡      (      a      +      b      +      c      )              sin      ⁡      a      +      sin      ⁡      b      +      sin      ⁡      c        &amp;lt;  1

Jayvion Tyler · Accepted Answer

sin  ⁡  (  a  )  +  sin  ⁡  (  b  )  &amp;gt;  sin  ⁡  (  a  +  b  ) if   (  a  ,  b  )  ∈  (  0  ,  π  )    ⟹    sin  ⁡  (  a  +  b  +  c  )  &amp;lt;=  sin  ⁡  (  a  )  +  sin  ⁡  (  b  +  c  )  &amp;lt;  sin  ⁡  (  a  )  +  sin  ⁡  (  b  )  +  sin  ⁡  (  c  )

pipantasi4 · Accepted Answer

Using  sin  ⁡  (  a  +  b  +  c  )  −  sin  ⁡  a  −  sin  ⁡  b  −  sin  ⁡  c  =  2  cos  ⁡      (                  2        a        +        b        +        c            2        )    sin  ⁡      (                  b        +        c            2        )    −  2  sin  ⁡      (                  b        +        c            2        )    cos  ⁡      (                  b        −        c            2        )  So  =  2  sin  ⁡      (                  b        +        c            2        )        [    cos    ⁡          (                        2          a          +          b          +          c                2            )        −    cos    ⁡          (                        b          −          c                2            )        ]    =  −  4  sin  ⁡      (                  a        +        b            2        )    sin  ⁡      (                  b        +        c            2        )    sin  ⁡      (                  a        +        c            2        )    &amp;lt;  0  ,Bcz given       a    ,    b    ,    c    ∈          (      0      ,              π        2            )      . So we get                     a        +        b            2        ,                  b        +        c            2            ,                  c        +        a            2        ∈          (      0      ,              π        2            )      So we get  sin  ⁡  (  a  +  b  +  c  )  &amp;lt;  sin  ⁡  a  +  sin  ⁡  b  +  sin  ⁡  c  ⇒            sin      ⁡      (      a      +      b      +      c      )              sin      ⁡      a      +      sin      ⁡      b      +      sin      ⁡      c        &amp;lt;  1

If a , b , c ∈ ( 0 , π<!-

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