1. When you say "a Dayquil Liquicap" do you mean ONE capsule ??
2. You should be aware that "determine" has a somewhat special meaning in analytical chemistry,i.e. the quantitation of the colorants in the capsule. Do you instead mean that you have been asked to identify the colorants ??
3. Your first task is to find out how many colorants are present; if you have only a single capsule, the simplest/quickest method might be paper chromatography (VERY small spot of the syrup applied at the centre of a filter paper/ dry thoroughly with COLD hair-dryer/ apply say about 10-20 uL of MeOH on top of dried spot/allow MeOH to migrate radially until dry/apply successive aliquots of MeOH, allowing to dry each time/ this may resolve the colorants into distinct bands). If this does not work well, you may have to move to proper paper chromatography in a glass tank or TLC or HPLC depending on equipment in your lab.
You might want to practice this technique with a non-valuable specimen e.g. first ball-point ink then a colored liquid like mouthwash, just to see how important the drying steps are.
Depending on the color of the capsule contents, and any colored bands separated, you may have FD&C Yellows or Reds (Yellow contents) or FD&C Blues or Greens (Green or Blue contents).
Research the UV spectra of any suspected FD&C's; a good place to look is the ALDRICH catalog. If you are lucky, the λmax values of the different FD&C's may be different enough that even in a complete syrup sample UV spectrum they (λmax) will be sufficiently resolved to identify the dyes present.
4. IF step 3 works well enough, you can purchase some FD&C dyes as food colorants at the local supermarket to use as standards to compare λmax values.
5. Note that some FDC dyes are sodium salts of polysulfonated azodyes and are not amenable to GC; if your particular dyes lack ionic groups, then GC may well be an option.
6. With only a single capsule, you will not be able to isolate enough pure material for NMR.
7. I believe that AA will tell you nothing, because of the complex mixture of excipients.
8. Stick with UV-vis until you have to resort to other techniques.
9. If you have multiple capsules, other options may open up, but generally these colorants are present at quite low levels and a LOT of work would be required for an undergraduate to isolate the purified colorants.
10. See step 8.
If you try step 3, please let us know how it works out.