The minimum phase difference between two SHM's is:
$y_1=\sin \left(\frac{\pi }{6}\right)\sin (\omega t)+\sin \left(\frac{\pi }{3}\right)\cos (\omega t)$

$y_2=\cos \left(\frac{\pi }{6}\right)\sin (\omega t)+\sin \left(\frac{\pi }{3}\right)\cos (\omega t)$ is:

$y_1=\sin (\omega t)\sin \left(\frac{\pi }{6}\right)+\cos$ $\omega t\sin \frac{\pi }{3}$

$=\sin$ $\omega t\cos \left(\frac{\pi }{3}\right)+\cos$ $\omega t\sin \frac{\pi }{3}$

$=\sin \left(\omega t+\frac{\pi }{3}\right)$

Similarly for the second equation is:
$y_2=\sin \left(\omega t+\frac{\pi }{6}\right)$

Thus phase difference between the two waves is 

$\frac{\pi }{3}-\frac{\pi }{6}=\frac{\pi }{6}$