We are given the parametric equations for the curve:

$x = t - \cos t \quad \text{and} \quad y = -1 + \sin t,$ and the question asks for the equation of the tangent line to the curve at $t = 0$.

Step 1: Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$

First, differentiate both parametric equations with respect to $t$.

For $x(t) = t - \cos t$: $\frac{dx}{dt} = 1 + \sin t.$

For $y(t) = -1 + \sin t$: $\frac{dy}{dt} = \cos t.$

Step 2: Slope of the tangent line

The slope of the tangent line at any point $t$ is given by the ratio of $\frac{dy}{dt}$ to $\frac{dx}{dt}$: $\text{slope} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\cos t}{1 + \sin t}.$

Now, substitute $t = 0$: $\frac{dy}{dt}\Big|_{t=0} = \cos(0) = 1,$ $\frac{dx}{dt}\Big|_{t=0} = 1 + \sin(0) = 1.$ Thus, the slope of the tangent line at $t = 0$ is: $\text{slope} = \frac{1}{1} = 1.$

Step 3: Find the point on the curve at $t = 0$

Substitute $t = 0$ into the parametric equations to get the coordinates of the point: $x(0) = 0 - \cos(0) = -1,$ $y(0) = -1 + \sin(0) = -1.$ So, the point of tangency is $(-1, -1)$.

Step 4: Equation of the tangent line

The equation of a line with slope $m$ passing through the point $(x_1, y_1)$ is: $y - y_1 = m(x - x_1).$ Substitute $m = 1$, $x_1 = -1$, and $y_1 = -1$: $y - (-1) = 1(x - (-1)),$ $y + 1 = x + 1,$ $y = x.$

Thus, the equation of the tangent line at $t = 0$ is: $y = x.$

Would you like more details or have any further questions?

Here are five related questions for further exploration:

1. How do we find the normal line to the curve at $t = 0$?
2. Can we generalize this process for finding tangents at other points $t$?
3. How would we interpret the curve if we eliminated the parameter $t$?
4. What are the applications of parametric curves in physics or engineering?
5. How do we use parametric equations to describe motion along a curve?

Tip: When solving parametric curve problems, always differentiate the parametric equations to find the slope, and evaluate carefully at the specific parameter value for accuracy.