For a party that loves to enforce the constitution as written, republicans have a hard time with the fourth amendment. Upon accepting their position, law enforcement agents swear to protect constitutional rights. No amount of training can nullify this oath or amendment. If we trash our freedoms, the terrorists win.
Of course the NYPD has the right to be apprised of any potential terrorist activity it hears of through regular channels. However, to go into Muslim mosques and other places of study and worship in an undercover way is just unconstitutional and makes for bad feelings between the police and all Muslims and free thinking people.
Look, its highly ignorant to assume that all sections of society produce an equal number of law breakers, The VAST majority here have no experience with law enforcement or peace keeping, so if people who are actually TRAINED to do so, feel the need to do so I don't think people should try and force their personal, and frankly ignorant convictions on them to try and make them change.