The best answer to this question depends on the definitions you're using for the trigonometric functions:
Unit circle:
t correspond to point (x,y) on the circle x^2+y^2 =1
Define:
sint = y, , cos t = x, , tant = y/x
The result is immediate.
Point (x,y) on the terminal side of t and r=sqrt(x^2+y^2)
sint = y/r, , cos t = x/r, , tant = y/x
sint/cost = (y/r)/(x/r)= (y/r)*(r/x) = y/x = tan t
Right triangle
sin theta = "opposite"/"hypotenuse" = "opp"/"hyp"
cos theta = "adjacent"/"hypotenuse" = "adj"/"hyp"
sin theta = "opposite"/"adjacent" = "opp"/"adj"
The result follow from :
sin theta / cos theta = ("opp"/"hyp")/("adj"/"hyp") = ("opp"/"hyp")*("hyp"/"adj") = "opp"/"adj" = tan theta
