No, nothing is obvious. In fact, there is no
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It appears that you are trying to express \(\displaystyle \sqrt{1 + \sqrt{i}}\) in \(\displaystyle a + bi\) form. (But I would not really call that "such an easy problem"!) Then you seem to be finding \(\displaystyle \sqrt{i}\) in exponential (polar) form.
Since addition doesn't work well in polar form, you might want to go back to rectangular ("general") form, as you almost did, in order to add, and then revert to polar form for the next root. What is \(\displaystyle \cos{\pi/4}\)?
There are other ways you might do it; I suppose your context (having been learning to use exponential form?) suggests the method you are using.