#x=cosalpha+isinalpha#,#y=cosbeta+isinbeta#,#z=cosgamma+isingamma# and#x+y+z=xyz#. prove that #cos(beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)=1#?

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Answer 1 · 2016-11-05T13:22:06

This proposition is false, e.g. when #alpha = 0#, #beta = pi/2#, #gamma=-pi/2#

Let:

#{ (alpha = 0), (beta = pi/2), (gamma = -pi/2) :}#

Then:

#{ (x = 1), (y = i), (z = -i) :}#

and:

#x + y + z = 1+i-i = 1 = 1*i*(-i) = xyz#

With these values, we find:

#cos(beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)#

#=cos(pi)+cos(-pi/2)+cos(-pi/2)#

#= -1+0+0=-1 != 1#