What methods are used to apprehend a pedophile?

I think you mean how do you apprehend a child sex offender. There is no actual law against paedophilia, which is a Greek word that describes someone who is attracted to pre-pubescent children (those aged 12 and under). No one can help how they think and feel, so the law couldn't apprehend someone just because they are a childlover (which is the English translation of the Greek word paedophile). To break the law, a paedophile must act on their attraction and molest a child (or at least have child porn in their possession). Most paedophiles learn how to live with their feelings and never touch a child. People who rape and murder a child, which is very rare, are not paedophiles at all, but homicidal maniacs who should be locked away. However, the media like to portray things as shockingly as possible. So the answer to your question is that you can't apprehend a paedophile just for being a paedophile. They have to first break the law. Even fifty or sixty years ago when homosexual acts committed between adults was a crime punishable by two years imprisonment, it wasn't actually illegal to be a homosexual, just to act on it.