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 \(\frac{dy}{dx} = \frac{r'(\theta) \sin(\theta) + r(\theta) \cos(\theta)}{r'(\theta) \cos(\theta) - r(\theta) \sin(\theta)}\) where \(r'(\theta)\) is the derivative of \(r(\theta)\) with respect to \(\theta\).

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

Step 4 ：Then, we can substitute \(\theta = \pi/2\) into the formula along with \(r(\theta)\) and \(r'(\theta)\) 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 \(\boxed{-1}\).