1. The problem is to simplify the expression $y^2$ for small values of $\theta$ using Taylor expansions. 2. Recall the Taylor expansion of cosine near zero: $$\cos \theta \approx 1 - \frac{\theta^2}{2}$$ 3. Substitute this approximation into the expression: $$y^2 \approx r^2 \left( \frac{5}{4} - \cos \theta \right) = r^2 \left( \frac{5}{4} - \left( 1 - \frac{\theta^2}{2} \right) \right)$$ 4. Simplify inside the parentheses: $$\frac{5}{4} - 1 + \frac{\theta^2}{2} = \frac{1}{4} + \frac{\theta^2}{2}$$ 5. Thus the expression becomes: $$y^2 = r^2 \left( \frac{1}{4} + \frac{\theta^2}{2} \right) = \frac{r^2}{4} + \frac{r^2 \theta^2}{2}$$ 6. This matches the required expression for $y^2$ when $\theta$ is small. Final simplified result: $$y^2 = \frac{r^2}{4} + \frac{r^2 \theta^2}{2}$$