Q98 / 200 · Manipal (MET) 2012 · Mathematics

$A+i B \text { form of } \frac{(\cos x+i \sin x)(\cos y+i \sin y)}{(\cot u+i)(1+i \tan v)}$is

  • Correct answer
    A
    $\sin u \cos v[\cos (x+y-u-v)$ $+i \sin (x+y-u-v)]$
  • B
    $\sin u \cos v[\cos (x+y+u+v)$ $+i \sin (x+y+u+v)]$
  • C
    $\sin u \cos v[\cos (x+y+u+v)$ $-i \sin (x+y+u+v)$
  • D
    None of the above

Solution

LHS $\begin{aligned} & =\frac{(\cos x+i \sin x)(\cos y+i \sin y)}{(\cos u+i \sin u)(\cos v+i \sin v)} \sin u \cos v \\ & =\sin u \cos v e^{i x} \cdot e^{i y} \cdot e^{-i u} \cdot e^{-i v} \\ & =\sin u \cos v[\cos (x+y-u-v) \\ & \quad+i \sin (x+y-u-v)] \end{aligned}$

Hence, the correct answer is option: 'A'.

More Complex Number Questions