Here is a rough outline of the proof. Fill in the blanks.
1. Suppose that then are both the product of an even number of transpositions. So argue that is also the product of an even number of transpositions.
2. Let be a transposition. What is ? What does that imply?
3. Let such that where are transpositions. Using 2. what is in terms of ? (if this doesn't make sense...try doing some small example and see the pattern)
I think you should be able to handle one. For the second one merely note that a transpotition is its own inverse. Therefore so that contains the identity.
If are are transpositions then . To see this merely note that . Therefore an even permutation has...........what?