Find the slope of the tangent to the curve $r=4-4\cos \theta$ at the value $\theta =\pi /2$

Step 1 ：We are given the polar equation $r=4-4\cos \theta$ and we are asked to find the slope of the tangent to the curve at the value $\theta =\pi /2$.

Step 2 ：The slope of the tangent to the curve in polar coordinates can be found using the formula $\frac{dy}{dx}=\frac{r^{\prime }(\theta )\sin (\theta )+r(\theta )\cos (\theta )}{r^{\prime }(\theta )\cos (\theta )-r(\theta )\sin (\theta )}$ where $r^{\prime }(\theta )$ is the derivative of $r(\theta )$ with respect to $\theta$.

Step 3 ：First, we need to find $r^{\prime }(\theta )$. Given $r(\theta )=4-4\cos (\theta )$, we can differentiate this with respect to $\theta$ to find $r^{\prime }(\theta )=4\sin (\theta )$.

Step 4 ：Then, we can substitute $\theta =\pi /2$ into the formula along with $r(\theta )$ and $r^{\prime }(\theta )$ to find the slope of the tangent at this point.

Step 5 ：The calculation gives us the slope of the tangent to the curve at the point $\theta =\pi /2$ to be -1. This means that the tangent line at this point is decreasing at a rate of 1 unit vertically for every 1 unit horizontally.

Step 6 ：Final Answer: The slope of the tangent to the curve $r=4-4\cos \theta$ at the value $\theta =\pi /2$ is $\boxed{{\displaystyle -1}}$.