What are the general procedures for simplifying a trigonometric expression using Euler's formula? As an example, this is how we can simplify the following trigonometric expression:
$$\sin{x}\cos{x}$$
$$\sin{x}\cos{x} = \dfrac{e^{ix}-e^{-ix}}{2i} \times \dfrac{e^{ix}+e^{-ix}}{2}$$
$$ = \dfrac{(e^{ix}-e^{-ix})(e^{ix}+e^{-ix})}{4i}$$ $$ = \dfrac{(e^{2ix}-e^{-2ix})}{4i}$$ $$ = \dfrac{1}{2} \times \dfrac{(e^{i(2x)}-e^{-i(2x)})}{2i}$$ $$ = \dfrac{1}{2} \times \sin(2x)$$
However, how would you approach other expressions? What general steps can you take to simplify any trigonometric expression, such as the following:
$$ \tan^{-1}(\dfrac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}})$$ Where $\dfrac{-\pi}{4} < x < \dfrac{\pi}{4}$
Replace the sines and cosines by the $e^{ix}$ expressions, and simplify. That'll make the argument to arctan a lot simpler. In fact, if you write $$ u = \tan^{-1}(\dfrac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}}) $$ then you can rewrite that as $$ \tan u = \dfrac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}} $$ The stuff on the right simplifies to some exponential involving $x$; the stuff on the left similarly simplifies to something involving $u$. Set these equal, and see whether you can find a relationship between $u$ and $x$ as a result.
You'll notice I haven't done the work for you --- that's because you asked "what general steps can be taken," and I've answered that.