Where did my simplification go wrong? Sum and difference formula simplification

Question

I'm struggling with the following: We are to use the sum and difference formulas to find the exact value of the expression. The problem is simplification has been tough. As a last resort I decided to use Symbolab to find the answer and steps but the steps were not to be found. Despite lack of steps, the answer is $(\sqrt {2+\sqrt{3}})/2$

Here are my steps so far: \begin{align*} \sin(135^\circ - 30^\circ) & = (\sin 135^\circ\cos 30^\circ)-(\cos 135^\circ \sin 30^\circ)\\ & = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)-\left(-\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)\\ & = \left(\frac{\sqrt{6}}{4}\right)-\left(-\frac{\sqrt{2}}{4}\right) \end{align*} [insert final simplification step]
$(\sqrt {2+\sqrt{3}})/2$

What's the missing step here?

Answer 1

\begin{align*} \frac{\sqrt{6}}{4}- \left(-\frac{\sqrt{2}}{4}\right) & = \frac{1}{4}(\sqrt6+\sqrt2)\\ & = \frac{\sqrt{2}}{4}(\sqrt{3}+1)\\ & = \frac{\sqrt{2}}{4}\sqrt{(\sqrt3+1)^2}\\ & = \frac{\sqrt{2}}{4}\sqrt{4+2\sqrt3}\\ & = \frac{\sqrt{2}}{4}\sqrt{2}\sqrt{2+\sqrt{3}}\\ & = \frac{1}{2}\sqrt{2+\sqrt{3}} \end{align*}

Answer 2

$x=\frac{\sqrt6+\sqrt2}4$
$====================$
$x^2=\frac{6+2\sqrt{12}+2}{16}=\frac{8+2\sqrt{12}}{16}=\frac{8+4\sqrt3}{16}=\frac{2+\sqrt3}4$
$====================$
$x=\frac{\sqrt{2+\sqrt3}}2$