Question:
let x,y,z∈R and such x+y+z=π,and such
tany+z−x4+tanx+z−y4+tanx+y−z4=1
show that
cosx+cosy+cosz=1
My idea: let x+y−z=a,x+z−y=b,y+z−x=c
then
a+b+c=π
and
tana4+tanb4+tanc4=1
we only prove
cosb+c2+cosa+c2+cosa+b2=1
Use
cosπ−x2=sinx2
⟺sina2+sinb2+sinc2=1
let
tana4=A,tanb4=B,tanπ4=C
then
A+B+C=1
and use sin2x=2tanx1+tan2x
so we only prove
2A1+A2+2B1+B2+2C1+C2=1
other idea:let
y+z−x4=a,x+z−y4=b,x+y−z4=c
then we have
a+b+c=π4,tana+tanb+tanc=1
we only prove
cos(2(b+c)+cos2(a+c)+cos2(a+b)=sin(2a)+sin(2b)+sin(2c)=1
then I fell very ugly, can you some can help?
Thank you very much!
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