Answer
the curve:
$$ r=\cos^{2}( \frac{\theta }{2}) $$
is completely traced with
$$
\theta =0 \quad \text {to} \quad \theta=2 \pi
$$
the exact length of the given curve is equal to
$$
\begin{aligned} L &=\int_{0}^{2 \pi} \sqrt{\cos ^{2}(\theta / 2)} d \theta=\\
&=\int_{0}^{2 \pi}|\cos (\theta / 2)| d \theta \\
&=4 \end{aligned}
$$
Work Step by Step
The curve:
$$ r=\cos^{2}( \frac{\theta }{2}) $$
is completely traced with
$$
\theta =0 \quad \text {to} \quad \theta=2 \pi
$$
is equal to
$$
\begin{aligned} r^{2}+(d r / d \theta)^{2} &=\left[\cos ^{2}(\theta / 2)\right]^{2}+\left[2 \cos (\theta / 2) \cdot(-\sin (\theta / 2)) \cdot \frac{1}{2}\right]^{2} \\ &=\cos ^{4}(\theta / 2)+\cos ^{2}(\theta / 2) \sin ^{2}(\theta / 2) \\ &=\cos ^{2}(\theta / 2)\left[\cos ^{2}(\theta / 2)+\sin ^{2}(\theta / 2)\right] \\ &=\cos ^{2}(\theta / 2) \end{aligned}
$$
then the exact length of the given curve is equal to
$$
\begin{aligned} L &=\int_{0}^{2 \pi} \sqrt{\cos ^{2}(\theta / 2)} d \theta=\\
&=\int_{0}^{2 \pi}|\cos (\theta / 2)| d \theta \\
&\quad\quad\quad[\text { since } \cos (\theta / 2) \geq 0 \text { for } 0 \leq \theta \leq \pi] \\
&=2 \int_{0}^{\pi} \cos (\theta / 2) d \theta \\
&\quad \quad\quad \left[u=\frac{1}{2} \theta\right] \\
&=4 \int_{0}^{\pi / 2} \cos u d u \\
&=4[\sin u]_{0}^{\pi / 2}\\
&=4(1-0)\\
&=4 \end{aligned}
$$
