Tuesday, February 7, 2017

trigonometry - Euler's formula simplification


I have been trying to simplify this expression:


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


into:


$$\ i \sin(\theta).$$


However I can't find what steps to take to get to this simplification.


Euler's formula states:


$$\ e^{i\theta}= \cos (\theta) + i \sin(\theta) $$


It is linked to this formula however I am not sure how to go about this.


Answer




$$cos(\theta)-\frac{1}{\cos(\theta)+i\sin(\theta)}=cos(\theta)-\frac{1}{e^{i\theta}}=$$ $$cos(\theta)-e^{-i\theta}=\frac{1}{2}e^{i\theta}+\frac{1}{2}e^{-i\theta}-e^{-i\theta}=$$ $$=\frac{1}{2}e^{i\theta}-\frac{1}{2}e^{-i\theta}=i\Big(\frac{1}{2i}e^{i\theta}-\frac{1}{2i}e^{-i\theta}\Big)=i\sin(\theta)$$


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