在不使用计算器的情况下，证明 $\sin 75^\circ = \frac{\sqrt{2} + \sqrt{6}}{4}$。

1. Express $75^\circ$ as a sum of special angles and apply the compound angle formula for sine.

$\sin 75^\circ = \sin(45^\circ + 30^\circ)$
$= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

2. Substitute the exact values of the trigonometric functions for $30^\circ$ and $45^\circ$.

$= \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$

3. Simplify the expression to obtain the required form.

$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$
$= \frac{\sqrt{2} + \sqrt{6}}{4}$