Is my understanding correct?
If a function is anti-symmetric about $T/2$ where $T$ is the period, then it posses half wave symmetry.
If a function has half-wave symmetry and symmetry about the midpoint of the positive and negative half-cycles, the periodic function is said to have quarter-wave symmetry.
By definition, does a cosine function have quarter-wave symmetry?
Also, does all odd functions have quarter-wave symmetery?