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

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