1. ## unusual equality

Find the solution of $e^z = z\text{ in }|z|<2$
and $\tan x = x\text{ in }-\frac{3}{2}\pi

2. Neither of those can be solved analytically. You will need a numerical method, such as graphing on a calculator.

For example, on the first one, you could set $y=e^x-x$ and then find all the zeros graphically on $-2.

3. I believe the first one is supposed to be a problem in complex analysis (use of $z$ rather than $x.)$ Perhaps Rouché’s theorem might help.

4. What is Rouche's theorem????
I have tried to solve this but it got fail >_<