Answer
$h=k(\tan B-\tan A)$
Work Step by Step
Name $x$ = the leg opposite to the angle $A$.
Using TOA in SOHCAHTOA to find x,
in the triangle where x is opposite to A:
$\displaystyle \tan A=\frac{x}{k}$
$ x=k\tan A$
in the triangle where (x+h) is opposite to B:
$\displaystyle \tan B=\frac{h+x}{k}$
$k\tan B=h+x$
$x=k\tan B-h$
Equate the two expressions for x and solve for h:
$k\tan A=k\tan B-h$
$h=k\tan B-k\tan A$
$h=k(\tan B-\tan A)$
