What expression represents DeMoivre's Theorem?

DeMoivre's Theorem provides a powerful formula for raising complex numbers to a power and relates to the polar form of complex numbers. The theorem states that if a complex number is represented in polar form as ( r(\cos x + i \sin x) ), then when raised to the power of ( n ), it can be expressed as ( (r^n)(\cos(nx) + i \sin(nx)) ).

In the expression representing DeMoivre's Theorem, the left side ( [r(\cos x + i \sin x)]^n ) shows the polar representation being raised to the ( n )-th power. On the right side, ( r^n(\cos(nx) + i \sin(nx)) ) demonstrates that the modulus ( r ) is raised to the power of ( n ), while the angle ( x ) is multiplied by ( n ). This aligns perfectly with the theorem's concept, confirming the correctness of the expression.

The other choices do not accurately represent DeMoivre's Theorem. For example, some options use incorrect forms or mix trigonometric identities improperly, while others misrepresent the powers of complex numbers or fail to maintain the relationship between polar and