The first and third identity are easy to prove. Recall that the modular discriminant $\Delta = \Delta(\omega_1, \omega_2)$ is defined by $\Delta = g_2^3 - 27g_3^2,$ where $g_2 = g_2(\omega_1, \omega_2) = 60G_4(\omega_1, \omega_2)$ and $g_3 = g_3(\omega_1, \omega_2) = 140G_6(\omega_1, \omega_2)$ are the Weierstrassian invariants. Moreover, we define the Klein invariant $J = J(\omega_1, \omega_2)$ and the $j$-function by $J = \frac{g_2^3}{\Delta} \quad \text{and} \quad j = 12^3 J,$ respectively.
Let $\lambda \ne 0$. Because $g_2$ and $g_3$ are of order $-4$ and $-6$, respectively, we have $g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2 \quad \text{and} \quad g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3,$ and $\Delta(\lambda \omega_1, \lambda \omega_2) = \lambda^{-12} \Delta,$ accordingly. Moreover, since $g_2^3$ and $\Delta$ are of the same order, $J(\lambda \omega_1, \lambda \omega_2) = J(\omega_1, \omega_2)$. In particular, $g_2(\tau) = \omega_1^4 g_2, \quad g_3(\tau) = \omega_1^6 g_3, \quad \text{and} \quad \Delta(\tau) = \omega_1^{12} \Delta$ On the other hand, $J = J(\tau) \quad \text{and} \quad j = j(\tau),$ that is to say, both $J$ and $j$ are functions of $\tau$ alone.
Using the classical identity $\Delta(\tau) = (2\pi)^{12} \eta^{24}(\tau)$, where $\eta(\tau)$ is the Dedekind eta function defined by $\eta(\tau) = q^{1/12} \prod_{n = 1}^\infty (1 - q^{2n}), \quad q = e^{\pi i \tau}, \quad \textbf{I}[\tau] > 0,$ we obtain $\eta^{24}(\tau) = \frac{\Delta(\tau)}{(2\pi)^{12}} = \frac{\omega_1^{12} g_2^3}{(2\pi)^{12} J(\tau)} = \frac{12^3 \omega_1^{12} g_2^3}{(2\pi)^{12} j(\tau)}.$ In view of a known result, $E_{2k}(q) = 1 - \frac{4k}{B_{2k}} \sum_{n = 1}^\infty \frac{n^{2k - 1} q^{2n}}{1 - q^{2n}}, \quad q = e^{\pi i \tau}, \quad \textbf{I}[\tau] > 0,$ where $B_{2k}$ are the Bernoulli numbers and $E_{2k}(q) = G_{2k}(\tau)/2\zeta(2k)$, we get $g_2(\tau) = 60G_4(\tau) = \frac{4\pi^4 E_4(q)}{3},$ or, what is the same thing, $g_2 = \omega_1^{-4} \frac{4\pi^4 E_4(q)}{3},$ Therefore, $\eta^{24}(\tau) = \frac{12^3 \omega_1^{12}}{(2\pi)^{12} j(\tau)} \left(\omega_1^{-4} \frac{4\pi^4 E_4(q)}{3}\right)^3 = \frac{E_4^3(q)}{j(\tau)}.$ Lastly, using the fact that $E_8(q) = E_4^2(q)$ leads to $\eta^{48}(\tau) = \frac{E_8^3(q)}{j^2(\tau)}.$
Now, the second identity follows from the first and the fact that $J(\tau) = \frac{E_4^3(q)}{E_4^3(q) - E_6^2(q)}.$ Indeed, we have $E_6^2(q) = E_4^3(q) - 12^3 \frac{E_4^3(q)}{j(\tau)},$ but in view of the first identity this becomes $E_6^2(q) = (j(\tau) - 12^3)\eta(\tau)^{24},$ and the result follows.