More mathematically, it can be done as :
The minute hand moves 360 degrees in 60 minutes. This means that the angle of the minute hand is given by 6t, where t is number of minutes past midnight.
The hour hand moves 30 degrees in 60 minutes. This means that the angle of the hours hand is given by 0.5t.
The hands start together at midnight. The first time they make a 90 degree angle is when the minute hand has moved 90 degrees further than the hour hand, so this is given by the equation:
6t = 0.5t + 90
5.5t = 90
t = 16 4/11 (16 minutes and 4/11 seconds)
In other words about 16 minutes past midnight.
The next time is when the minutes hand has gained another 180 degrees on the hour hand, and is 90 degrees behind it:
6t = 0.5t + 270
5.5t = 270
t = 49 1/11 (49 minutes and 1/11 seconds)
At about 11 minutes to 1 o'clock.
For every 180 degrees that the minute hand gains on the hour hand there will be one 90 degree angle, so every 49 1/11 - 16 4/11 = 32 8/11 minutes
24 hours is 1440 minutes.
1440/(32 8/11 ) = 44
So every 24 hours there are 44 right angles between minute hand and second hand.
Hope it helps :)