How is the is this equation for the height of a triangle derived from sine law?

I don't think that formula is quite correct.
The sine rule, as I expect you know, is

 .. a ........... b ......... c
 ------- .. = --------- = ----------
 sin A ..... sin B ..... sin C

If you draw the height h perpendicular to side b, then from the rightangled triangle on one side of it you get
 h = c sin A, and from the sine rule we see that

 .... a sin C .....b sin C
c = ------------ = ------------
 ...... sin A ....... sin B

Replacing c by one of these fractions in the expression for h gives
 
 h = a sin C in the first case
and

 ......b sin C sin A
 h = -------------------- in the second case.
 .........sin B

Drawing the perpendicular to the other sides would give other heights, but the formulas for them would be

 ......c sin B sin A
 h = --------------------
 .........sin C

and

 ......a sin B sin C
 h = --------------------
 .........sin A