Find the exact value of cos pi/12 using half angle identities

[tex]\bf cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1+cos({{ \theta}})}{2}}\\\\ -------------------------------\\\\ \cfrac{\pi }{12}\cdot 2\implies \cfrac{\pi }{6}\qquad therefore\qquad \cfrac{\quad \frac{\pi }{6}\quad }{2}\implies \cfrac{\pi }{12}\qquad then \\\\\\ cos\left( \frac{\pi }{12} \right)\implies cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\sqrt{\cfrac{1+cos\left( \frac{\pi }{6} \right)}{2}}[/tex]

[tex]\bf cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\sqrt{\cfrac{1+\frac{\sqrt{3}}{2}}{2}}\implies cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\sqrt{\cfrac{\frac{2+\sqrt{3}}{2}}{2}} \\\\\\ cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\sqrt{\cfrac{2+\sqrt{3}}{4}}\implies cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\cfrac{\sqrt{2+\sqrt{3}}}{\sqrt{4}} \\\\\\ cos\left( \cfrac{\frac{\pi }{6}}{2} \right)=\pm\cfrac{\sqrt{2+\sqrt{3}}}{2}[/tex]