We know that ,
#color(red)((1)csc^2theta-cot^2theta=1=>csc^2theta=1+cot^2theta#
#color(blue)((2)cos^2theta+sin^2theta=1#
We take LHS :
#LHS=(1+(cscAtanB)^2)/(1+(cscCtanB)^2)#
#color(white)(LHS)=(1+color(red)(csc^2A)color(brown)(tan^2B))/(1+color(red)(csc^2C)color(brown)(tan^2B))tocolor(red)(Apply(1)#
#color(white)(LHS)=(1+(color(red)(1+cot^2A))color(brown)(sin^2B/cos^2B))/(1+
(color(red)(1+cot^2C))color(brown)(sin^2B/cos^2B))to[because color(brown)(tantheta=sintheta/costheta)]#
#color(white)(LHS)=
(color(blue)(cos^2B+sin^2B)+cot^2Asin^2B)/(color(blue)(cos^2B+sin^2B)+cot^2Csin^2B
)tocolor(blue)( Apply(2)#
#color(white)(LHS)=(color(blue)(1)+cot^2Asin^2B)/(color(blue)(1)+cot^2Csin^2B)#
#color(white)(LHS)=(1+(cotAsinB)^2)/(1+(cotCsinB)^2#
#LHS=RHS#