Answer
$$
x=t-2 \sin t, \quad y=1-2 \cos t, \quad 0 \leq t \leq 4 \pi
$$
the length of the given curve is
$$
\begin{aligned}
L &=\int_{a}^{b} \sqrt{(d x / d t)^{2}+(d y / d t)^{2}} d t \\
&=\int_{0}^{4 \pi} \sqrt{5-4 \cos t} d t \\
& \approx 26.7298
\end{aligned}
$$
Work Step by Step
$$
x=t-2 \sin t, \quad y=1-2 \cos t, \quad 0 \leq t \leq 4 \pi
$$
then
$$
d x / d t=1-2 \cos t , \quad d y / d t=2 \sin t,
$$
so
$$
\begin{aligned}
(d x / d t)^{2}+(d y / d t)^{2}& =(1-2 \cos t)^{2}+(2 \sin t)^{2} \\
&=1-4 \cos t+4 \cos ^{2} t+4 \sin ^{2} t \\
&=5-4 \cos t
\end{aligned}
$$
Thus the length of the given curve is
$$
\begin{aligned}
L &=\int_{a}^{b} \sqrt{(d x / d t)^{2}+(d y / d t)^{2}} d t \\
&=\int_{0}^{4 \pi} \sqrt{5-4 \cos t} d t \\
& \approx 26.7298
\end{aligned}
$$
