Am I improving my tactical knowledge by explaining solutons to my wife? I recently started to teach the game to my wife, yay for me :) So now she's somewhat comfortable with how the pieces move and the basic opening principles but as soon as she gets to the middle game she never knows what to do next so I started showing her tactical puzzles, we are doing the book by Fred Reinfeld "1001 Winning Chess Sacrifices and Combinations".

Since she is not experienced once I've set up the position I see the solution and explain to her the tactical weaknesses in order to point her in the direction of the solution. She still needs some time to find it but eventually she gets it.

My question is, since I am not finding the solution by myself it would seem that I am not benefitting from these exercises but since I explain to her in detail that in a particular position there is a weak pawn, or an overload piece or a way to force a pin could this be helping me as well or the only way to improve tactical knowledge is by solving the puzzles, period?
Since you are teaching your wife, who's new at chess, these tactics are perhaps a bit too easy for you. They probably won't do much for your standard game. 

That said, by explaining the patterns to her and doing a lot of 'easier' tactics, you are definitely helping your blitz/bullet games. 

Basically you'll be able to see tactics faster in your games, but your ability to see more complicated tactics is remaining more or less unchanged. To really hone your tactics, you're going to need to improve your ability to calculate deep, and consider several different lines when perhaps the solution isn't as obvious.  

Teaching is great for reminding yourself of the basics, but if you want to improve, you're going to need to do more than teach. Find some more difficult tactics for yourself, play some challenging games, study more opening lines, and analyze your shortcomings. 

Hope this answers your question.