Monday, 20 May 2013

trigonometry - How prove this cosx+cosy+cosz=1

Question:




let x,y,zR and such x+y+z=π,and such
tany+zx4+tanx+zy4+tanx+yz4=1
show that
cosx+cosy+cosz=1





My idea: let x+yz=a,x+zy=b,y+zx=c
then
a+b+c=π
and
tana4+tanb4+tanc4=1
we only prove
cosb+c2+cosa+c2+cosa+b2=1
Use
cosπx2=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+zx4=a,x+zy4=b,x+yz4=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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