Question

If $y=\cos \theta +i\sin \theta$, then the value of $y+1/y$ is equals to ?

• A. $2\cos \theta$
• B. $2\sin \theta$
• C. $2\cos ec\theta$
• D. $2\tan \theta$

Answer 1

The correct option is A

$2\cos \theta$

Explanation for the correct option:

Finding the value of $y+\frac{1}{y}$:

Given function,

$y\mathit{}\mathit{=}\mathit{}\cos \mathit{}\theta \mathit{}+\mathrm{i}\mathit{}\sin \mathit{}\theta \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}...\left(1\right)$

Then $\frac{1}{y}=\frac{1}{\cos \theta +i\sin \theta }$

Multiply numerator and denominator by$\cos \theta -i\sin \theta$,

$\begin{array}{rcl}\frac{1}{y}& =& \frac{(\cos \theta -i\sin \theta )}{\left(\cos \theta +i\sin \theta \right)\left(\cos \theta -i\sin \theta \right)}\\ & =& \frac{\left(\cos \theta -i\sin \theta \right)}{({\cos }^2\theta +{\sin }^2\theta )}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{}\left(a+b\right)\left(a-b\right)\mathbf{=}{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}\mathbf{}\mathbf{,}{\mathbf{i}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{]}\mathbf{}\\ & =& \cos \mathrm{\theta }-\mathrm{i}\sin \mathrm{\theta }...\left(2\right)\end{array}$

By adding $\left(1\right)$ and $\left(2\right)$, we get

$\begin{array}{rcl}y+\frac{1}{y}& =& \cos \theta +i\sin \theta +\cos \theta -i\sin \theta \\ & =& 2\cos \theta \end{array}$

Hence, the correct option is A.