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Let us start from one side of what we want to prove,$$\eta(-1/\tau) = r^{1/24} \prod_{n = 1}^\infty (1 - r^n), \quad r = \exp\left(-{{2\pi i}\over\tau}\right).$$We were given Euler's (quite remarkable) pentagonal number theorem, so we might as well start by using it to transform the product into a sum\eta(-1/\tau) = r^{1/24} \sum_{n \in \mathbb{Z}} (-1)^n ...
Since I am not so much familiar with the theory of modular forms, I am offering a solution which uses theory of theta functions and their link with elliptic integrals. With this approach the functional relation of $y(\tau)$ is transformed into a modular equation of degree $3$ which can be independently verified using the theory of modular equations. Let \$\...