Equivalence of chordal and spherical metric on Riemann Sphere

Question

The chordal metric between two points $z,w \in \hat{\mathbb{C}} (=\mathbb{C}\cup{\{\infty\}})$ is defined as , \begin{align*} d(z,w)=\displaystyle\frac{2|z-w|}{\sqrt{(1+|z|^2)(1+|w|^2)}} \end{align*} \begin{align*} d(z,\infty)=\displaystyle\frac{2}{\sqrt{(1+|z|^2)}} \end{align*} The spherical metric is defined as goes , $\gamma \colon [0,1] \to \mathbb{C}\cup{\{\infty\}}$; The spherical length of the curve $\gamma$ in $ \hat{\mathbb{C}} $ is defined as, \begin{align*} \Lambda(\gamma)= \int_\gamma ds = \int_\gamma \displaystyle\frac{2|dz|}{1+|z|^2} \end{align*} The spherical distance between two points is the infimum of the spherical lengths of paths joining the two points: \begin{align*} \rho (z,w) := inf_\gamma \Lambda(\gamma) \end{align*} $\gamma$ varies over all paths joining $z$ and $w$. In "Normal Families" by schiff, it's written that , \begin{align*} d(z,w) \leq \rho(z,w) \leq \dfrac{\pi}{2} d(z,w) \end{align*} I cannot figure out why is this true .