Why square side from its diagonal does not equal $\frac{\sqrt{2}}{2}$?

Question

If diagonal of square is known, we can consider the square as two triangles.

We know hypotenuse of the triangles and all of angles ($45^\circ$, $45^\circ$, $90^\circ$).

So, as in picture above, if we know AC, we can find AD, for example, as

\begin{align} \sin 45^\circ &= \frac{AD}{AC} \\ \frac{\sqrt{2}}{2} &= \frac{AD}{AC} \\ AD &= AC\frac{\sqrt{2}}{2} \end{align}

However, the formula given is $$AD = \frac{AC}{\sqrt{2}} .$$ What am I missing?

Answer 1

$\frac{AC}{\sqrt{2}} = AC\frac{1}{\sqrt{2}} = AC(\frac{\sqrt{2}(1)}{\sqrt{2}\sqrt{2}}) = AC\frac{\sqrt{2}}{2}$

Answer 2

$$\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{\sqrt{2}\sqrt{2}}=\frac{1}{\sqrt{2}},$$ so it's just the same