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Explicit formulas are proved for the $5$-torsion points on the Tate normalform $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$.These formulas express the coordinates of points in $E_5[5] -\langle(0,0)\rangle$ as products of linear fractional quantities in terms of$5$-th roots of unity and a parameter $u$, where the parameter $b$ whichdefines the curve $E_5$ is given as $b=(\varepsilon^5 u^5-\varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(\tau)$ isthe Rogers-Ramanujan continued fraction and $b=r^5(\tau)$, then the coordinatesof points of order $5$ in $E_5[5] - \langle(0,0)\rangle$ are shown to beproducts of linear fractional expressions in $r(5\tau)$ with coefficients in$\mathbb{Q}(\zeta_5)$.

Product formulas for the $5$-division points on the Tate normal form and the Rogers-Ramanujan continued fraction