Solve both parts and justify the conditions used. Part A: Are arguments θ and θ+2π different complex numbers at the same positive modulus? Explain. Part B: Evaluate the power.

1. Part A reasoning

2. Use modulus and argument, remembering that arguments differ by full turns.

3. Calculate or simplify this relation.

   $$r(\cos(\theta+2\pi)+i\sin(\theta+2\pi))=r(\cos\theta+i\sin\theta)$$

4. A representation is not unique even though the complex number is fixed.

5. Part B reasoning

6. Expand with i²=−1 and use a conjugate to remove a complex denominator.

7. Calculate or simplify this relation.

   $$i^{42}=(i^4)^{10}i^2=-1$$

8. Reduce the exponent modulo four.

A: No; sine and cosine are unchanged by one full turn. B: The requested value is -1.

Checks and common pitfalls: A representation is not unique even though the complex number is fixed. Reduce the exponent modulo four.