Let's consider the following expression:
$(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} + \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$
The left hand side of the equality represents only one value. On the other hand, the right hand side represents an infinitely number of periodic values, one of which is $\cos(15\sqrt{2}^\circ)$. But not only the cosine of this irrational angle is a value of the right hand side, but also is the principal value.
In fact, you can check that both expressions are numerically equal when evaluated in Wolfram Alpha.
The value of the lect-hand side is here
http://m.wolframalpha.com/input/?i=cos+%282%5E1%2F2pi%2F12%29&x=0&y=0
And the value of the right hand side of $(1)$
The question is: Is correct the use of the equality sign between a single-valued function and a multiple-valued one, by the convention that the multi-valued function takes his principal value? is there any objection to the equality $(1)$ or is it correct by the convention of the principal value?