Prove that

L.H.S

$=2 \cos 45^{\circ} \cos 15^{\circ}$

$=2 \cos 45^{\circ} \cos \left(45^{\circ}-30^{\circ}\right)$

$=2 \frac{1}{\sqrt{2}}\left(\cos 45^{\circ} \cos 30^{\circ}+\sin 45^{\circ} \sin 30^{\circ}\right)$

$=\sqrt{2}\left(\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \times \frac{1}{2}\right)$

$=\sqrt{2}\left(\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}\right)$

$=\sqrt{2}\left(\frac{\sqrt{3+1}}{2 \sqrt{2}}\right)$

$=\frac{\sqrt{3}+1}{\sqrt{2}}$