When analysing the motion of the particle we should resolve vertically and horizontally.
Resolving vertically we obtain
(1)
Resolving horizontally, we have to use the fact that the acceleration for a particle moving in a circle is of magnitude
or
and directed towards the centre of the circle. We apply
(2)
(2) divided by (1) gives
Then
Unlike as in simple harmonic motion the period does depend on the size of the motion.