TWO SMALL AIRCRAFT HAVE A NMAC AT CPS; THE LOCAL CONTROLLER DID NOT PROACTIVELY TELL BOTH AIRCRAFT ABOUT EACH OTHER FOR VISUAL IDENTIFICATION.

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DURING A TRAFFIC PATTERN TRAINING LESSON AT CPS; WITH 3 TOTAL PERSONS ON BOARD; WE WERE CLEARED TO LAND ON RUNWAY 30L. WHILE ON THE DOWNWIND LEG ON SHORT FINAL; TOWER INSTRUCTED US TO GO AROUND AND STEP TO THE LEFT. AS WE DID THIS; ANOTHER AIRCRAFT; WHAT LOOKED LIKE A HIGH WINGED AIRCRAFT; PASSED BENEATH US AND TO THE RIGHT AT ABOUT A 200 FOOT RANGE. AFTER THIS; WE WERE INSTRUCTED TO CIRCLE AND PROCEEDED TO MAKE A NORMAL; SAFE LANDING; WHICH WE DID. OF THE 3 OCCUPANTS IN THE AIRCRAFT; NONE OF US HEARD THE OTHER PLANE CLEARED TO LAND. I PULLED UP THE TAPES OF THE TOWER LATER ON; AND HEARD A BROKEN TRANSMISSION AND ANOTHER PLANE CLEARED TO LAND ON RUNWAY 30L. THERE WERE NO TRAFFIC ADVISORIES GIVEN TO US IN ANY WAY OR FORM. TOWER HAD CLEARED 2 PLANES TO LAND AT THE SAME TIME ON THE SAME RUNWAY WITHOUT ANY ADVISORIES TO EITHER AIRCRAFT. I WAS CONTINUOUSLY SCANNING FOR TRAFFIC; BUT DID NOT SEE THE OTHER PLANE APPROACHING THE AIRPORT. I DO NOT BELIEVE ANY PHYSIOLOGICAL FACTORS TO BE INVOLVED. I WAS NOT FATIGUED; AND ONLY MILDLY STRESSED (AS ANY CFI IS). THE SITUATION WAS NOT ESCALATED IN ANY WAY AFTER THIS; AND WE SAFELY LANDED AND WENT BACK TO THE FBO. SUPPLEMENTAL INFO FROM ACN 815431: THE FLIGHT WAS FROM ZZZ TO CPS; LANDING IN CPS AROUND XA25. (LOCAL SUNSET WAS AT XA39.) THE SETTING SUN MADE FORWARD VISIBILITY DIFFICULT DESPITE THE GOOD VFR CONDITIONS. AS SUCH; I WAS UNABLE TO SEE THE AIRPORT FROM MY WESTERLY HEADING UNTIL I WAS APPROX 5 OR 6 MILES OUT; AT WHICH TIME I WAS CLEARED FOR A VISUAL APPROACH TO RUNWAY 30L; FOR WHICH I WAS ALREADY ON A RIGHT BASE; AND TURNED OVER TO THE TOWER APPROX 4-5 MILES FROM THE AIRPORT. I SWITCHED TO THE TOWER AND WAS CLEARED TO LAND. I DID NOT HEAR ANY OTHER AIRCRAFT ON THE FREQUENCY NOR WAS I INFORMED OF ANY CONFLICTING TRAFFIC BY THE CONTROLLER. AFTER I ROLLED OUT ON FINAL; I HEARD THE TOWER CONTROLLER TELL ANOTHER AIRCRAFT TO 'GO AROUND AND CLIMB.' I IMMEDIATELY LOOKED IN FRONT OF ME; ASSUMING THAT I MUST HAVE BEEN OVERTAKING AN AIRCRAFT ON FINAL BUT SAW NOTHING ON FINAL OR OVER THE RUNWAY. AT THAT POINT; I LOOKED OVER MY LEFT SHOULDER AND SAW THE OTHER AIRCRAFT BELOW AND SLIGHTLY BEHIND ME CLIMBING AND BANKING AWAY FROM ME. AT THAT POINT I WOULD ESTIMATE THE AIRPLANE WAS LESS THAN 100 FT HORIZONTALLY FROM ME AND CLIMBING THROUGH MY ALTITUDE. I AM UNABLE TO SAY HOW CLOSE WE WERE WHEN THE CONTROLLER GAVE THE GO-AROUND INSTRUCTION. I AM NOT SURE WHETHER THE OTHER PILOT HAD SEEN ME PRIOR TO THE CONTROLLER'S INSTRUCTION. I CONTINUED MY APPROACH AND LANDED NORMALLY. THE CONTROLLER GAVE NO INDICATION THAT EITHER AIRCRAFT HAD FAILED TO FOLLOW AN INSTRUCTION NOR OF WANTING TO PURSUE THE ISSUE FURTHER. I BELIEVE THIS INCIDENT COULD HAVE BEEN PREVENTED HAD THE CONTROLLER BEEN MORE PROACTIVE WITH REGARD TO SEPARATING THE 2 AIRCRAFT. A TRAFFIC ADVISORY WOULD HAVE PROVIDED ME WITH THE OPPORTUNITY TO ADJUST MY FLIGHT PATH TO FIT IN BEHIND THE OTHER AIRCRAFT. CONSIDERING THE LIGHTING CONDITIONS AT THE TIME OF THE INCIDENT; INSTRUCTIONS FROM THE TOWER WOULD HAVE BEEN EVEN MORE HELPFUL.