Although I completely agree with Turamarth's response, I want to point out some aspects of "why" they can not copy secret/hidden techniques.
A Sharingan user can copy all types of jutsus techniques: ninjutsu, genjutsu and taijutsu.
The examples are:
- Ninjutsu, we can see it in Kakashi who is an expert.
- Gentjutsu, we can see it in Itachi copying it from Kurenai.
- Taijutsu, the best example of this is Sasuke using the Lee taijutsu.
In other hand, we have the special exceptions that are:
Kekkei Genkai / Tota / Mora: This can't be copied cause this aren't techniques, this are a genetic abilities.(1)
Hidden jutsus: These techniques also cannot be copied through any means of simple observation, but this techniques can be taught to anyone. A few hidden techniques require a special ability or lineage and as such are usually pride and fame of the clan that possess them. Also, the most of these techniques are based on a Yin-Yang manipulation and this isn't a trivial knowledge.(2)
Summoning Jutsus: These techniques first need to sign a contract with the owner blood for to be able to do it.
Nature Transformations: They can't mimic a jutsu with a nature transformation that they haven't learned to perform.(3)
(1) Dust Release techniques these techniques are not genetic, as Onoki stated that Mu passed the secrets of Dust Release on to him. (Naruto chapter 525)
(2) For special abilities or lineage it mean some special characteristic of some one who use these techniques, like the big size of the people of the Akimichi clan.
(3) Nature transformations are Fire, Wind, Earth, Water, Lightning, and the other derivatives.
[A] So the answer is no, with only the Sharingan they can't copy hidden jutsus.