If a = cos(alpha) +i sin(alpha), b = cos(beta) +i sin(beta), c= cos(gamma) + i sin(gamma) and a+b+c = 0, then what is the value of a-1 +b-1+c-1?

Given:cosα+cosβ+cosγ+isinα+sinβ+sinγ=0+i0cosα+cosβ+cosγ=0   ---(1)and, sinα+sinβ+sinγ=0   ---(2)a=cosα+isinαalso, a¯=cosα-isinαa-1=1cosα+isinαmultplying by a¯ in numerator and denominator:a-1=cosα-isinαa2  (since a·a=a2)a-1=cosα-isinαcos2α+sin2α=cosα-isinα   (since cos2α+sin2α)Similarly, we will have:b-1=cosβ-isinβand,c-1=cosγ-isinγTherefore:a-1+b-1+c-1=cosα+cosβ+cosγ-isinα+sinβ+sinγUsing (1) and (2):a-1+b-1+c-1==0

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