通过将 $105^\circ$ 写成两个角之和，在不使用计算器的情况下证明：
$\sin 105^\circ = \frac{\sqrt{2}}{4}(\sqrt{3} + 1).$ 由此，用其参考角表示 $105^\circ$，并求 $\cos 165^\circ$ 的精确值。

1. Express $105^\circ$ as the sum of two special angles

$\sin 105^\circ = \sin(60^\circ + 45^\circ)$

2. Apply the sine addition formula

$\sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ$

3. Substitute exact values for the trigonometric ratios of special angles

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

4. Simplify the expression to obtain the required form

$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{2}\sqrt{3} + \sqrt{2}}{4} = \frac{\sqrt{2}}{4}(\sqrt{3} + 1)$

1. Find the basic angle of $105^\circ$ and relate it to $\sin 105^\circ$.

The basic angle of $105^\circ$ is $180^\circ - 105^\circ = 75^\circ$.
Therefore, $\sin 105^\circ = \sin(180^\circ - 75^\circ) = \sin 75^\circ$.
From part (i), $\sin 75^\circ = \frac{\sqrt{2}}{4}(\sqrt{3} + 1)$.

2. Express $\cos 165^\circ$ in terms of the basic angle $75^\circ$ and evaluate.

We can write $165^\circ$ as $90^\circ + 75^\circ$.
Using the addition formula for cosine:
$\cos 165^\circ = \cos(90^\circ + 75^\circ)$
$\cos 165^\circ = \cos 90^\circ \cos 75^\circ - \sin 90^\circ \sin 75^\circ$
Since $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$:
$\cos 165^\circ = -\sin 75^\circ$
$\cos 165^\circ = -\frac{\sqrt{2}}{4}(\sqrt{3} + 1)$