# If $(a_1+ib_1)(a_2+ib_2)......................(a_n+ib_n)=c+id$ then what is the value of $tan^{-1}\large\frac{b_1}{a_1}$+$tan^{-1}\large\frac{b_2}{a_2}$+............................$tan^{-1}\large\frac{b_n}{a_n}$?

(A) $n\pi-tan^{-1}\large(\frac{d}{c})$ (B) $n\pi+tan^{-1}\large(\frac{d}{c})$ (C) $2n\pi-tan^{-1}\large(\frac{d}{c})$ (D) $2n\pi+tan^{-1}\large(\frac{d}{c})$

Toolbox:
• $If\:\: tanx=tany$  then  $x=n\pi+y$  $\forall n\in N$
Let $a_k=r_kcos\theta_k$ and $b_k=r_ksin\theta_k$
where $k=1,2,...........n$
$\Rightarrow\:tan^{-1}\large(\frac{b_k}{a_k})=\theta_k$
Also let $c=Rcos\theta$ and $d=Rsin\theta$
$\Rightarrow\:tan^{-1}\large(\frac{d}{c})=\theta$
$(a_1+ib_1)(a_2+ib_2)............(a_n+ib_n)=c+id$
$\Rightarrow\:r_1(cos\theta_1+isin\theta_1)r_2(cos\theta_2+isin\theta_2)........r_n(cos\theta_n+isin\theta_n)$.
$=R(cos\theta+isin\theta)$.
$\Rightarrow\:r_1r_2.......r_n[cos(\theta_1+\theta_2+......\theta_n)+isin(\theta_1+\theta_2+......\theta_n)]$
$=R(cos\theta+isin\theta)$
comparing the arguments on both the sides,
$\theta_1+\theta_2+............\theta_n=n\pi+\theta$