1. Transformar a senos y cosenos:

• De la primera ec: $\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = x \implies \sin \theta \cos \theta = \frac{1}{x}$
• De la segunda ec: $\frac{1}{\cos \theta} + \frac{1}{\sin \theta} = \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} = y \implies \sin \theta + \cos \theta = y(\sin \theta \cos \theta) = \frac{y}{x}$

2. Eliminar la variable:
Elevamos al cuadrado la relación de la suma:
$$(\sin \theta + \cos \theta)^2 = \left(\frac{y}{x}\right)^2$$
$$\sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = \frac{y^2}{x^2}$$
$$1 + 2\left(\frac{1}{x}\right) = \frac{y^2}{x^2}$$

Multiplicamos todo por $x^2$:
$$x^2 + 2x = y^2$$