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Question
If to ω ≠ 1 is a cube root of unity, then show that `("a" + "b"omega + "c"omega^2)/("b" + "c"omega + "a"omega^2) + ("a" + "b"omega + "c"omega^2)/("c" + "a"omega + "a"omega^2)` = – 1
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Solution
L.H.S
`("a" + "b"omega + "c"omega^2)/("b" + "c"omega + "a"omega^2) + ("a" + "b"omega + "c"omega^2)/("c" + "a"omega + "a"omega^2)`
= `(("a" + "b"omega + "c"omega^2)/("b" + "c"omega + "a"omega^2)) omega/omega + (("a" + "b"omega + "c"omega^2)/("c" + "a"omega + "b"omega^2)) omega^2/omega^2`
= `(("a" + "b"omega + "c"omega^2)omega)/("b"omega + "c"omega^2 + "a"omega^3) + (("a" + "b"omega + "c"omega^2)omega^2)/("c"omega^2 + "a"omega^3 + "b"omega^4)`
= `(("a" + "b"omega + "c"omega^2)omega)/(("b"omega + "c"omega^2 + "a")) + (("a" + "b"omega + "c"omega^2)omega^2)/(("c"omega^2 + "a" + "b"omega))` ......`[because omega^3 = 1, omega^4 = omega^3 * omega = omega]`
= `omega + omega^2` ......`[because 1 + omega + omega^2 = 0]`
= – 1
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