We know that,
#color(red)((1)sin^-1(-x)=-sin^-1x and cos^-1(-x)=pi-cos^-1x#
#color(blue)((2)cos^-1x=tan^-1(sqrt(1-x^2)/x) #
#color(blue)((3)sin^-1x=tan^-1(x/sqrt(1-x^2))#
#color(violet)((4)tan^-1x-tan^-1y=tan^-1((x-y)/(1+xy))#
#color(green)((5)tan^-1(-x)=-tan^-1x#
#color(orange)((6)tan(-theta)=-tantheta#
#color(brown)((7)tan(pi+theta)=tantheta#
#(8)tan(tan^-1x)=x ,AA x in RR#
Let,
#X=tan(sin^-1(-12/13)-cos^-1(-3/5))...tocolor(red)(Apply(1)#
#=tan(-sin^-1(12/13)-(pi-cos^-1(3/5))#
#=tan(-sin^-1(12/13)-pi+cos^-1(3/5))#
#=tan(-pi+cos^-1(3/5)-sin^-1(12/13)).tocolor(blue)(Apply(2),
(3)#
#=tan(-pi+tan^-1(sqrt(1-9/25)/(3/5))-tan^-1((12/13)/sqrt(1-144/169)))#
#=tan(-pi+tan^-1(4/3)-tan^-1(12/5))...tocolor(violet)(Apply(4)#
#=tan(-pi+tan^-1((4/3-12/5)/(1+(4/3)(12/5))))#
#=tan(-pi+tan^-1((20-36)/(15+48)))#
#=tan(-pi+tan^-1(-16/63))...tocolor(green)(Apply(5)#
#=tan(-pi-tan^-1(16/63))#
#=tan(-(pi+tan^-1(16/63))...tocolor(orange)(Apply(6)#
#=-tan(pi+tan^-1(16/63))...tocolor(brown)(Apply(7)#
#=-tan(tan^-1(16/63)...toApply(8)#
#=-16/63#
#~~-0.2540#
