If : tan(x +iy) = a + bi then find a,b


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Question Number 96290 by 175 last updated on 31/May/20

$I f : \tan (x + i y) = a + b i$ $t h e n f i n d a, b$
Commented by Tony Lin last updated on 31/May/20

$t a n (x + i y)$ $= \frac{t a n x + t a n i y}{1 - t a n x t a n i y}$ $= \frac{t a n x + i t a n h y}{1 - t a n x \cdot i t a n h y}$ $= \frac{(t a n x - t a n h y i) (1 + t a n x t a n h y i)}{(1 - t a n x t a n h y i) (1 + t a n x t a n h y i)}$ $= \frac{t a n x + {t a n x t a n h}^{2} y}{1 + {t a n}^{2} {x t a n h}^{2} y} + \frac{{t a n}^{2} x t a n h y - t a n h y}{1 + {t a n}^{2} {x t a n h}^{2} y} i$
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