Which of the following represents a parabola?

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(B) For option $(A)$: $x=4 \cos t, y=4 \sin t$. Squaring and adding,$x^2+y^2=16(\cos^2 t + \sin^2 t) = 16$. This represents a circle.For option $(B)$:

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$x^2-2=-2 \cos t, y=\cos^2\left(\frac{t}{2}\right)$

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(B) For option $(A)$: $x=4 \cos t, y=4 \sin t$. Squaring and adding,$x^2+y^2=16(\cos^2 t + \sin^2 t) = 16$. This represents a circle.For option $(B)$: $x^2-2=-2 \cos t$ and $y=\cos^2\left(\frac{t}{2}\right)$.Using the identity $\cos t = 2\cos^2\left(\frac{t}{2}\right)-1$,we have $y = \frac{1+\cos t}{2}$,which implies $\cos t = 2y-1$.Substituting this into the first equation: $x^2-2 = -2(2y-1) = -4y+2$.Thus,$x^2 = -4y+4$,which is $x^2 = -4(y-1)$. This is the equation of a parabola.

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