Question

If (cos$\theta$+isin$\theta$)(cos2$\theta$+isin2$\theta$)

(cosn$\theta$+isinn$\theta$)=1, then the value of $\theta$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

We have (cos$\theta$+isin$\theta$)(cos2$\theta$+isin2$\theta$)

..........(cosn$\theta$+isinn$\theta$)=1

$\Rightarrow$ (cos($\theta$+2$\theta$+3$\theta$+.......n$\theta$)+isin($\theta$+2$\theta$+3$\theta$+.......n$\theta$)=1

$\Rightarrow$ cos$\left(\frac{n(n+1)}{2}\theta \right)$ + isin$\left(\frac{n(n+1)}{2}\theta \right)$=1

cos$\left(\frac{n(n+1)}{2}\theta \right)$=1 and sin$\left(\frac{n(n+1)}{2}\theta \right)$=0

$\Rightarrow$ $\frac{n(n+1)}{2}$$\theta$ = 2m$\pi$ $\Rightarrow$ $\theta$= $\frac{4m\pi }{n(n+1)}$, Where m $\in$I