Represntation of a complex number in polar form

Question

Question:

The answer given in the textbook is option d. What if I take iota outside the bracket which gives

(iota)^4 (cos theta + iota*sin theta)^4

1*(cos 4theta + iota*sin 4theta)

Which means option c. What am I doing wrong?

Answer 1

Then it must be $$i^4\left(\frac{\sin(\theta)}{i}+\cos(\theta)\right)^4$$

Answer 2

$$\sin \theta + i\cos \theta= i(\cos \theta -i \sin \theta)$$ so that $$(\sin \theta + i\cos \theta)^4= i^4(\cos \theta -i \sin \theta)^4= (\cos\theta -i \sin\theta)^4 = \cos 4\theta - i\sin 4\theta$$