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Question
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`
Solution
Let (x1 + iy1) (x2 + iy2) (x3 + iy3) …… (xn + iyn) = a + ib
Taking arguments on both sides
arg[(x1 + iy1) (x2 + iy2)(x3 + iy3) …… (xn + iyn)] = arg(a + ib)
arg(x1 + iy1) + arg(x2 + iy2) + arg(x3 + iy3) …… + arg(xn + iyn) = arg(a + ib)
`tan^-1 (y_1/x_1) + tan^-1 (y_2/x_2) + tan-1 (y_3/x_3) + ... + tan^-1 (y_"n"/x_"n") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`
`sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`
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