Which of the following is not a valid polar form representation of a complex number as given?

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In polar form a complex number with magnitude r and angle θ is written as z = r(cos θ + i sin θ). Expanding this gives z = r cos θ + i r sin θ, so the same factor r multiplies both the real part and the imaginary part.

Option A matches exactly this pattern, and option C is also valid because cos θ − i sin θ = cos(−θ) + i sin(−θ), so z = r(cos θ − i sin θ) is just the same polar form with angle −θ.

The remaining forms fail for general r because the imaginary part must be multiplied by r as well. In z = r cos θ − i sin θ, the real part has the r factor but the imaginary part does not, so you cannot factor out a single r to get the standard r(cos φ + i sin φ). Similarly, z = r cos θ + i sin θ lacks the r factor in the imaginary part, which again cannot be written as a single r multiplied by (cos φ + i sin φ) unless r = 1.

So the form that is not a valid polar form for general r is the one where the imaginary part is not scaled by r.