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Sign of Trigonometric Ratios in Different Quadrants

Let fz = si...

Question

Let $f (z) = s i n z$ and $g (z) = c o s z$ . If * denotes a composition of function then show that $(f + i g) * (f - i g) = i e^{- e^{i z}}$ .

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Solution

$(f - i g) (z) = f (z) - i g (z) = s i n z - i c o s z = - i (c o s z + s i n z) = - i e^{i z}$
$∴ (f + i g) * (f - i g) (z) = (f + i g) (- i e^{i z}) = f (- i e^{i z}) + i g (- i e^{i z})$
$= s i n (- i e^{i z}) + i c o s (- e^{i z})$
$= i e^{- i (- i e^{i z})} = i e^{- e^{i z}}$

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