Reading some Conformal Field Theory, I came across the following equation about the Jacobi Theta functions without any justification:

Let $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$$ and $$\theta_{3}(q)=\sum_{n \in \mathbb{Z}}q^{n^{2}}$$ then:

$$\frac{\theta_{3}(q^{2})\theta_{3}(\overline{q}^{2})+\theta_{2}(q^{2})\theta_{2}(\overline{q}^{2})}{\theta_{3}(q)\theta_{3}(\overline{q})}=\frac{1}{\sqrt{2}}\sqrt{\left|\frac{\theta_{2}^{4}(q)}{\theta_{3}^{4}(q)}\right|+1+\left|1-\frac{\theta_{2}^{4}(q)}{\theta_{3}^{4}(q)}\right|}.$$

Indeed, this seems to hold true when I plot the two sides. Does somebody has a clue why this equation holds ? Any comments or any reference ?

Thanks !