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The Temple of Trajan has not actually been excavated, hence the ambiguity surrounding its fate. However, it is thought to have been at where the 16th century Santa Maria di Loreto is presently located. It was not unusual for Medieval or Renaissance churches to be built on ancient temples.
Excavations of the [Loreto] foundations disclosed a number of large fragments of marble, probably from the Temple of the Deified Trajan; and Michelangelo may have carved one of the larger pieces into the base for the equestrian statue of Marcus Aurelius.
Packer, James E. The Forum of Trajan in Rome: A Study of the Monuments in Brief. University of California Press, 2001.
Construction on the Loreto Church began in 1507, though it was known to have been built over an earlier chapel. This implies that the Temple of Trajan was destroyed some time in the 15th century, probably by having a chapel built over its foundations and its marble recycled.
In all likelihood, the temple was in a state of extreme disrepair before its final destruction.