Cos a =5/13 with a in quadrant IV Cos b = -12/13 with b in quadrant 11 Solve tan(a-b)?

tan a = - 12/5
tan b = - 5/12

tan a - tan b
-----------------
1 + tan a tan b

-12/5 + 5/12
---------------------
1 + 1

-144 + 25
-------------
120

- 119 / 120

** a in Q IV => sin a < 0. sin^2a + cos^2a = 1 =>
sin^2a = 1 - ( 5/13)^2 = 144/169 => sin a = +/- 12/13. We choose sin a = - 12/13 , then tan a = sina/cosa = ( - 12/13)/(5/13) = - 12/5
** b in Q II => sin b > 0 . sin^b + cos^2b = 1 =>
sin^b = 1 - ( - 12/13 )^2 = > sinb = 5/13 ; then tanb = sinb/cosb = (5/13)/ (- 12/13) = - 5/12
** tan ( a - b ) = ( tan a - tan b )/ ( 1 + tan a tan b ) = [ - 12/5 - ( - 5/12 )] / [ 1 + ( - 12/5 )( - 5/12 )] =
[ ( - 144 + 25)/ 60 ] / 2 = - 119/ 120