Question

If $x+iy=\dfrac{1}{(1+cosθ)+isinθ}$,then the value of $x^2+1$ is ?

• A. $\dfrac{7}{4}$
• B. $\dfrac{13}{4}$
• C. $\dfrac{1}{4}$
• D. $\dfrac{9}{4}$
• E. $\dfrac{5}{4}$

Answer 1

Answer: (E)

$\dfrac{5}{4}$

Answer 2

Given that:

$x+iy=1/(1+cosθ)+isinθ$

$=\dfrac{1}{1+2cos^2(θ/2)+i.2sin(θ/2).cos(θ/2)}$

$=\dfrac{1}{1+2cos^2(θ/2)}.\dfrac{1}{e^{i.(θ/2)}}$

$=\dfrac{1}{1+2cos^2(θ/2)}.e^{-i.(θ/2)}$

$=\dfrac{1}{1+2cos^2(θ/2)}.(cos(θ/2)-isin(θ/2))$

$=\dfrac{1}{2}-itan(θ/2)$

Hence, now we can write that $x=\dfrac{1}{2}$

So, $x^2+1=(\dfrac{1}{2})^2+1=\dfrac{5}{4}$ (_Ans.)