1. Datos y fórmulas:
Sea $\theta = 7.5^{\circ}$. Buscamos $E = \tan \theta + \cot \theta$.
Sabemos que $\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}$.
Usando el ángulo doble: $\sin 2\theta = 2 \sin \theta \cos \theta$, entonces $E = \frac{2}{\sin 2\theta}$.

2. Desarrollo paso a paso:
Calculamos $2\theta$:
$$ 2\theta = 2(7.5^{\circ}) = 15^{\circ} $$
El valor de $\sin 15^{\circ}$ se puede obtener por diferencia de ángulos:
$$ \sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ}\cos 30^{\circ} - \cos 45^{\circ}\sin 30^{\circ} = \frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\frac{1}{2} = \frac{\sqrt{6}-\sqrt{2}}{4} $$
Sustituimos en $E$:
$$ E = \frac{2}{\frac{\sqrt{6}-\sqrt{2}}{4}} = \frac{8}{\sqrt{6}-\sqrt{2}} $$
Racionalizamos multiplicando por $\sqrt{6}+\sqrt{2}$:
$$ E = \frac{8(\sqrt{6}+\sqrt{2})}{6-2} = \frac{8(\sqrt{6}+\sqrt{2})}{4} = 2(\sqrt{6}+\sqrt{2}) $$

3. Conclusión:
El valor simplificado es $2(\sqrt{6}+\sqrt{2})$.
$$ \boxed{2(\sqrt{6}+\sqrt{2})} $$