Why exactly is tan x = sin x / cos x

Calculus question

In a right angled triangle, we have
...|.\
A |...\ C
...|_x_\
.....B

tan x = A/B
sin x = A/C
cos x = B/C
=> tan x = sinx / cosx = (A/C)/(B/C) = A/B right from the definition !

tan = opposite/adjacent
sin = opposite/hypotenuse
cos = adjacent/hypotenuse

(opposite/hypotenuse) / (adjacent/hypotenuse) = (opposite/hypotenuse) * (hypotenuse/adjacent) = opposite/adjacent

given a right triangle
sin a = Opp/ hyp
cos a = adj / hyp
tan a = Opp/ adj

sin a / cos a = (Opp/ hyp)/(adj/hyp) = Opp / adj = tan a