<s>[INST] Use an appropriate Half-Angle Formula to find the exact value of $\sin 105^\circ$ Is this correct? $$\sin 105^\circ = \frac{\sqrt{2+\sqrt{3}}}{4}$$
Because $105^\circ$ is in the second quadrant the square root after the equal sign is $+$. However, because $\cos 210^\circ$ is negative, it value is negative  $-\frac{\sqrt{3}}{2}$. 
Is my thought process correct? 
Also, can I factor out $\frac{1}{4}$? 
 [/INST] $$
\sin(105^\circ)
= \sin(90^\circ + 15^\circ)
= \cos(15^\circ)
= \sqrt{ \frac{1 + \cos(30^\circ)}{2}}
= \sqrt{ \frac{1 + \frac{1}{2} \sqrt{3}}{2}}
= \sqrt{ \frac{2 + \sqrt{3}}{4}}.
$$
</s>