Key phrase is hexagonal crystals in urinary sediment which means means its Cystine (i remember it as SIX-tine for hexagon) crystals which is made up of 2 cysteine molecules.

Cysteine Urinary Stones:

Hexagonal Crystals

Positive Nitroprusside test

Due to a hereditary inability to reabsorb : Cysteine, Ornithine, Lysine, And Arginine ( COLA ) ----> Only Cysteine percipitates to make renal stones although all can be found in the Urine.

The diagnosis can be made based on the Characteristic test they mentioned, The Hexagonal shape of the stones, and the fact that his brother has a similar condition ( points to hereditary )

Tyrosine and copper dont cause kidney stones

although calcium is very common it has a different shape and isn't gonna be positive for nitroprusside test.

FA 2017: Urinary cyanide-nitroprusside test is diagnostic. not sure how to know that it was a red color, but maybe just the fact that red seemed like a positive test result.

link to more background but not necessarily helpful for this question