Question #3dd7c

1 Answer
Feb 16, 2018

=-2csc2xcot2x

Explanation:

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