which of the following equations would transform the tangent graph to the parent cotangent graph?\n$y =…

Explanation:

Step1: Recall trigonometric identities

We know that $\cot x=\tan\left(x + \frac{\pi}{2}\right)$.

Step2: Analyze each option

For $y = \tan\left(x+\frac{\pi}{2}\right)$, it directly transforms the tangent - graph to the cotangent graph according to the identity $\cot x=\tan\left(x+\frac{\pi}{2}\right)$. For $y=-\tan(x - \pi)=-\tan x$ (using the periodicity of the tangent function $\tan(x + k\pi)=\tan x,k\in\mathbb{Z}$), it is just a reflection of $y = \tan x$ about the $x$ - axis and a shift that does not transform it to the cotangent graph. For $y=-\tan\left(-x-\frac{\pi}{2}\right)=\tan\left(x+\frac{\pi}{2}\right)$ (using the odd - function property of the tangent function $\tan(-\alpha)=-\tan\alpha$), but the negative sign in front changes its orientation compared to the direct transformation. For $y =-\tan\left(x-\frac{\pi}{2}\right)=\cot x$ (using the identity $\tan\left(x-\frac{\pi}{2}\right)=-\cot x$), but with the negative sign it is a reflection of the cotangent graph.

Answer:

$y=\tan\left(x+\frac{\pi}{2}\right)$