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The expression ((cos x+isin x)(cos y+isin y))/((cot u+i)(1+itan v)) can be written in the form A+iB. Which of the following is it equal to?

1. sin ucos v [cos(x+y-u-v)+isin(x+y-u-v)]
2. sin ucos v [cos(x+y+u-v)+isin(x+y+u-v)]
3. sin ucos v [cos(x+y+u+v)-isin(x+y+u+v)]
4. None of these

Correct answer: sin ucos v [cos(x+y-u-v)+isin(x+y-u-v)]

Solution

Numerator = e^(i(x+y)). cot u + i = e^(iu)/sin u and 1 + i tan v = e^(iv)/cos v, so denominator = e^(i(u+v))/(sin u cos v). Dividing gives sin u cos v * e^(i(x+y-u-v)) = sin u cos v[cos(x+y-u-v) + i sin(x+y-u-v)].

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