1) The cosine function is periodic with period and thus , hence The conclusion follows.
2) If is even then for some and hence (which follows from above). Notice that Thus if is even then the statement is true since Note that due to the cosine function's periodic attributes . If is odd then for some and hence Notice that Thus if is odd then the statement is true since hence the statement is true
3) I have to go so I will leave this one until later unless of course someone else would like to fill in the blanks.