First let's solve #sintheta + csctheta = 2#
Using #csctheta=1/sintheta# and getting common denominators the equation becomes
#sin^2theta/sintheta+1/sintheta=2#
Multiplying both sides by #sintheta# to remove the fraction, and rearranging, the resulting equation is
#sin^2theta-2sintheta+1=0#
This quadratic equation factors to #(sintheta-1)(sintheta-1)=0#
Using the zero product rule,
#sintheta-1=0#
#sintheta=1#
#theta=pi/2# for #0<=theta<=2pi#
Now, #sectheta=1/costheta# and #cos(pi/2)=0#
So #sec(pi/2)=1/0# which is undefined.
so #sec^3(pi/2)=(1/0)^3# which is also undefined.
#csc(pi/2)=1# since #sin(pi/2)=1# and #csc(pi/2)=1/sin(pi/2)#
so #csc^3theta=csc^3 (pi/2)=1#
So since the solution for #0<=theta<=2pi# to the equation
#sintheta+csctheta=2# is #theta=pi/2#, the value of #sec^3theta +csc^3theta# is "undefined" + 1, or undefined.
