This is a doubt of complex no. Please revert back as soon as possible.

Note that i^4=1 

So the denominator can be multiplied by i. As a result, the denominator now becomes the conjugate of the numerator. Now simply use Euler's form and write the numerator as re^ix and since the denominator is the conjugate, it will be written as re^-ix. So their division would result in e^i2x= cos2x+isin2x. But what is x? x is the argument of the numerator. But tanx=sintheta/(1+costheta)= tan(theta/2).

So that x=theta/2

Now raising the LHS to the power of 4, we get 4 theta as the argument on LHS, wall the argument on RHS is n theta. Hence n is equal to 4.