<s>[INST] Where did my simplification go wrong? Sum and difference formula simplification 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?
 [/INST] $x=\frac{\sqrt6+\sqrt2}4$
$====================$
$x^2=\frac{6+2\sqrt{12}+2}{16}=\frac{8+2\sqrt{12}}{16}=\frac{8+4\sqrt3}{16}=\frac{2+\sqrt3}4$
$====================$
$x=\frac{\sqrt{2+\sqrt3}}2$    
</s>