Why, according to Wolfram Alpha, is $y\approx0.206$ a solution to $e^{iy}+e^{-iy}=8ye^{4y^2}$ but not to $e^{iy}\cdot\left(e^{iy}+e^{-iy}\right)=e^{iy}\cdot\left(8ye^{4y^2}\right)$?
Surely since I've multiplied both sides of the equation by a constant term, the solution should remain the same? Why doesn't it? Is there a mistake in my methods or in WA?
Thanks.
There seems to be a mistake in your methods, since this root is indeed solution of both equations. Actually all solutions WA shows are valid for both of the equations. It is just confusing since WA shows only some roots and in this case they happen to be different for each equation, but still valid.