How is time averaged cos²(Φ/2) is ½ ?

i.e., how is cos²(Φ/2)> is ½ where angular brackets represent time averaging?

If averaged over the complete period, it would also give zero as the time averages cos²(Φ/2).

But if solved for half the period, or using the even function property of the function, would be resolved to 1+cos(Φ) and integrated for half the period only. This result when averaged over the period over which it was integrated would result in ½.