How many times are the hands of a clock at $90$ degrees. 
How many times are the hands of a clock at right angle in a day? 

Initially, I worked this out to be $2$ times every hour. The answer came to $48$.
However, in the cases of $3$ o'clock and $9$ o'clock, right angles happen only once.
So the answer came out to be $44$.
Is the approach correct?

Yes, but a more “mathematical” approach might be this: In a 12 hour period, the minute hand makes 12 revolutions while the hour hand makes one. If you switch to a rotating coordinate system in which the hour hand stands still, then the minute hand makes only 11 revolutions, and so it is at right angles with the hour hand 22 times. In a 24 hour day you get 2×22=44.
