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.
