What is the mistake? $$1=1$$
$$\Rightarrow\frac{-1}{1}=\frac{1}{-1}$$
$$\Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$
$$\Rightarrow\frac{i}{1}=\frac{1}{i}$$
$$\Rightarrow\frac{i}{2}=\frac{1}{2i}$$
$$\Rightarrow\frac{i}{2}+\frac{3}{2i} = \frac{1}{2i} +\frac{3}{2i}$$
$$\Rightarrow i(\frac{i}{2}+\frac{3}{2i} ) =  i(\frac{1}{2i} +\frac{3}{2i})$$
$$\Rightarrow\frac{-1}{2}=\frac{1}{2}$$
$$\Rightarrow1=2$$
What is wrong in this?

The trick is assuming that
$$\sqrt{\frac{-1}{1}} = \sqrt{\frac{1}{-1}}\Rightarrow 
\frac{\sqrt{-1}}{\sqrt{1}} = \frac{\sqrt{1}}{\sqrt{-1}}$$
which is not true because by definition of (complex) square root $-1=\sqrt{-1}\cdot \sqrt{-1} \not=\sqrt{1}\cdot \sqrt{1}=1$.
On the other hand, note that if $a,b,c,d$ are real POSITIVE numbers then 
$$\sqrt{\frac{a}{b}}=\sqrt{\frac{c}{d}}\Leftrightarrow 
\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{c}}{\sqrt{d}}.$$
