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?

\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*}
