If $z=\cos \theta + i \sin \theta$, express $\displaystyle \frac {1}{1-z \cos \theta}$ in the form $a+i\cdot b$.

Question

In order to get rid of the $z$, should I substitute $z=\cos\theta + i\sin \theta$ into the complex number, or what conjugate should I multiply the complex number by?

I have tried substituting $z=\cos\theta + i\sin \theta$ into the complex number, but only got this far:

$\displaystyle \frac {1}{1-z \cos \theta}$

$= \displaystyle\frac{1}{1-(\cos\theta + i\sin\theta)(\cos\theta)}$

$=\displaystyle\frac{1}{1-\cos^2\theta-i\cos\theta\sin\theta}$

As for multiplying the complex number by a conjugate, I have used $\big(\displaystyle\frac {1}{z}-\frac{1}{\cos\theta}\big)$, $\big(\displaystyle z-\frac{1}{z}\big)$ and $(1+z\cos\theta)$ but to no avail.

I have only learnt de Moivre's theorem, and I haven't learnt $\cosθ+i\sinθ=e^{iθ}$, so I would appreciate if this question can be solved in the simplest way possible. But other methods are welcome.

Answer 1

$$\dfrac1{1-\cos\theta(\cos\theta+i\sin\theta)}=\dfrac1{\sin^2\theta-i\sin\theta\cos\theta}=\dfrac1{-i\sin\theta}\cdot\dfrac1{\cos\theta+i\sin\theta}=\dfrac{\cos\theta-i\sin\theta}{-i\sin\theta}$$

Answer 2

\begin{align} \frac1{1-z\cos\theta}&=\frac1{1-(\cos\theta+i\sin\theta)\cos\theta}\\ &=\frac1{1-\cos^2\theta-i\sin\theta\cos\theta}\\ &=\frac1{\sin^2\theta-i\sin\theta\cos\theta}\color{blue}{\cdot\frac{\sin^2\theta+i\sin\theta\cos\theta}{\sin^2\theta+i\sin\theta\cos\theta}}\\ &=\frac{\sin^2\theta+i\sin\theta\cos\theta}{\sin^4\theta+\sin^2\theta\cos^2\theta}\\ &=\frac{\sin^2\theta+i\sin\theta\cos\theta}{\sin^2\theta(\sin^2\theta+\cos^2\theta)}\\ &=\frac{\sin^2\theta}{\sin^2\theta}+\frac{i\sin\theta\cos\theta}{\sin^2\theta}\\ &=1+i\cot\theta \end{align}

Answer 3

Note that $z=e^{i\theta}$, $z=e^{-i\theta}$ and $\cos\theta = \frac{z+\bar z}2$

$$\frac {1}{1-z \cos \theta}=\frac {1}{1-z \cos \theta}\cdot\frac {1-\bar z \cos \theta}{1-\bar z \cos \theta}$$ $$=\frac {1-e^{-i\theta}\cos \theta}{1-(\bar z +z)\cos \theta + \cos^2\theta}$$ $$=\frac {1-(\cos\theta - i\sin\theta)\cos \theta}{1-2\cos\theta\cos \theta + \cos^2\theta}$$ $$=\frac {1-\cos^2\theta + i\sin\theta\cos \theta}{1- \cos^2\theta}$$ $$=1+ i\cot\theta$$