Skill synergy was a thing in 3.5 - not so much in Pathfinder. The best one can do is to roll each perform individually for the purpose of income checks taking the best result only.
GMs would have to judge if second attempts could be made with related knowledge or craft/profession checks. Usually, adventure paths will only allow one check per PC, if a PC lacks the knowledge or craft skill that is most applicable - a related skill might be used to substitute at a greater DC check. Synergy does not exist in pathfinder.
If you want to house rule skill synergy, related skills added a +2 bonus, limit of one bonus per check. (Example, diplomacy adding to knowledge:local when seeking information).
You can try the Influence mechanic from Ultimate Intrigue.
http://www.d20pfsrd.com/gamemastering/other-rules/intrigue/#Influence
Perhaps you want a Discovery phase and then an Influence phase (where the Discovery phase provides some bonus to the Influence check), or perhaps you treat it like multiple characters doing an Aid Another as part of either check.
Successful Discovery checks give +4 to other Discovery checks, and informs you of which skill has the easiest Influence DC.
Influence has this mechanic (that you generally use when you have to meet multiple checks; they aren't usually optional in this mechanic):
"Additional checks serve as aid another attempts tied to the principal check. Succeeding at an influence check by a substantial margin provides additional benefits. Succeeding at an influence check by 5 or more counts as succeeding at an influence check and a discovery check (the PC chooses whether to learn one of the skills that influences the NPC, one of the NPC’s strengths, or one of the NPC’s weaknesses after the check is rolled instead of before the check, but the check otherwise functions as a successful discovery check). Succeeding at an influence check by 10 or more allows the PC to choose between gaining the benefit of succeeding at two influence checks or the benefits of an influence check and a discovery check (as if she had succeeded by only 5 or more)."
I just allow character to make 2 checks and take the best result (like KastrelZ suggested). Or, in cases like "DC 30 perception or DC15 Kn Engineering to notice sabotage in the mechanism" if any of them succeeds.
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Several things stop the bard in question from doing that:
1) it is not possible to help yourself. You need a second character using the aid another action.
That is why in RL we have bands.
2) If you allow rolling for each somewhat applicable skill and keep the better result, probably an untrained character singing, banging a bongo and dancing at the same time would get similar results to those that a skilled character with the same charisma would get. Rolling 3 dices and keeping the better has a good chance of getting better results that rolling a single dice with a +4 (1 rank and class skill).
The reaction of some player to the use of a home rule that allow to roll all the potentially relevant skills will be to try to always use several skills. Obnoxious players shouldn't be encouraged.
Generally using the skill with the better modifier encompass the use of the other relevant skills.
The GM can decide to add a small bonus, but usually it should be lesser than what you get from the help of other characters, or it will make the other characters redundant.
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The GM can decide to add a small bonus, but usually it should be lesser than what you get from the help of other characters, or it will make the other characters redundant.
This is generally how I treat it. For the (Strings and Sing) bard, I'd allow him to pick the secondary skill and roll for a 1/2 strength aid another [DC 10 for +1]. So getting backup singers would net you double the bonus per performer.
I might allow a full aid another IF we're not talking simultaneous skill checks: for instance, if someone makes a craft/profession [paintings/painter] to make the set/props for a perform [act], I'd allow the full +2.