The crystals identified in a cloudy straw-colored fluid from a patient's big toe that appeared needle-shaped are:

The identification of needle-shaped crystals in a cloudy straw-colored fluid from a patient's big toe points to monosodium urate crystals, which are typically associated with gout. Gout is a form of inflammatory arthritis characterized by the accumulation of uric acid in the joints. When uric acid levels in the body increase, it can lead to the formation of monosodium urate crystals that precipitate in synovial fluid, particularly during a gout attack. These crystals are often observed under polarized light microscopy, appearing as needle-shaped and exhibiting favorable birefringence.

Cholesterol crystals tend to form in conditions such as cholesterol embolization or certain types of joint disease but usually appear as flat, rectangular or rhomboid shapes rather than needles. Hyaluronidase is an enzyme that breaks down hyaluronic acid but does not form crystals. Calcium pyrophosphate crystals, on the other hand, are associated with pseudogout and are typically rhomboid-shaped rather than needle-like. The characteristics of the crystals found in this scenario align specifically with monosodium urate, thus making it the correct identification.