Let
f(x)=csc2x
f(x+Deltax)=csc2(x+Deltax)
f(x+Deltax)-f(x)=csc2(x+Deltax)-csc2x
Now,
lim((f(x+Deltax)-f(x))/((x+Deltax)-Deltax))=(csc2(x+Deltax)-csc2x)/(Deltax)
=1/(Deltax)((csc2(x+Deltax)-csc2x)/(Deltax))
=1/(Deltax)(1/sin(2(x+Deltax))-1/sin(2x))
=1/(Deltax)((sin2x-sin2(x+Deltax))/(sin(2(x+Deltax))sin2x))
SinC-sinD=2cos((C+D)/2)sin((C-D)/2)
implies
C=2x, D=2(x+Deltax)
(C+D)/2=(2x+2(x+Deltax))/2
=(2x+2x+2Deltax)/2
=(4x+2Deltax)/2
=2(2x+Deltax)/2
(C+D)/2=2x+Deltax
(C-D)/2=(2x-2(x+Deltax))/2
=(2x-2x-2Deltax)/2
=(-2Deltax)/2
(C-D)/2=-Deltax
sin2x-sin2(x+Deltax)=2cos(2x+Deltax)sin(-Deltax)
lim(Deltaxto0)((f(x+Deltax)-f(x))/((x+Deltax)-Deltax))=1/(Deltax)(2cos(2x+Deltax)sin(-Deltax))/(sin(2(x+Deltax))sin2x)
=(2)(-sin(Deltax)/(Deltax))(1/sin(2x))((cos(2x+Deltax))/(sin(2(x+Deltax))))
(-2)/sinxlim(Deltaxto0)(sin(Deltax)/(Deltax)) lim(Deltaxto0)((cos(2x+Deltax))/(sin(2(x+Deltax))))
lim(Deltaxto0)(sin(Deltax)/(Deltax))=1
Now,
=-2cscx(1)(cos2x)/sin(2x)
=-2csc2xcot2x