How would I prove $\frac{1}z=\operatorname{cis}(-\theta)$ given $z=\operatorname{cis}(\theta)$?

Question

I know $\operatorname{cis}(-\theta)$ equals $\cos(\theta)-i\sin(\theta)$ and $\frac{1}z=\frac{1}\cos+i\frac{1}\sin=\sec+\frac{\csc}i$ (please, correct me if I'm wrong.)

I just don't know how to relate these two now. Can anyone provide some guidance?

Answer 1

assuming you mean $\text{cis}(\theta)=e^{\theta i}=\cos\theta+i\sin\theta$ you can just write $$\text{cis}(-\theta)=e^{-\theta i}=\cos(-\theta)+i\sin(-\theta)=\cos\theta-i\sin\theta\\ \frac{1}{z}=\frac{1}{\cos\theta+i\sin\theta}\\ =\frac{1}{\cos\theta+i\sin\theta}\cdot\frac{\cos\theta-i\sin\theta}{\cos\theta-i\sin\theta}\\ =\frac{\cos\theta-i\sin\theta}{\cos^2\theta-(i\sin\theta)^2}\\ =\frac{\cos\theta-i\sin\theta}{\cos^2\theta+\sin^2\theta}=\cos\theta-i\sin\theta=\text{cis}(-\theta)$$ where you just multiply by the conjugate to eliminate the complex number on below part, and use the facts that $i^2=-1$ and $\cos^2\theta+\sin^2\theta=1.$

P.s.: the step you took

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

is wrong because you can't generally say that $\frac{1}{a+bi}=\frac{1}{a}+\frac{i}{b}.$

Answer 2

It is not correct in general to say if $z=a+bi$ then $\frac 1 z = \frac 1 a + \frac 1 b i; $

rather, $\frac 1 z = \frac a {a^2+b^2} - \frac b {a^2+b^2}i. $

Note that $\cos^2\theta + \sin^2\theta = 1$.

Therefore, if $z=e^{i\theta}=\cos \theta + i \sin \theta$ then $\frac 1 z =e^{-i\theta}=\cos\theta-i\sin\theta.$