Before we begin the conversion, please observe that the secant function has a division by zero issue at #theta = pi/2 + npi#. The same is true for the cosecant function and #theta = npi#. This translates into the Cartesian as a restriction of #x !=0 and y != 0#
For #csc(theta)#, begin with:
#y = rsin(theta)#
#1/sin(theta) = r/y#
#1/sin(theta) = sqrt(x^2 + y^2)/y#
#csc(theta) = sqrt(x^2 + y^2)/y#
A similar substitution exists for the secant function:
#sec(theta) = sqrt(x^2 + y^2)/x#
Substitute #x^2 + y^2# for #r^2# and #sqrt(x^2 + y^2)# for r:
#x^2 + y^2 + sqrt(x^2 + y^2) = 2theta - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x != 0 and y!=0#
The substitution for #theta# breaks the equation into 3 equations:
#x^2 + y^2 + sqrt(x^2 + y^2) = 2tan^-1(y/x) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x > 0 and y>0#
#x^2 + y^2 + sqrt(x^2 + y^2) = 2(tan^-1(y/x) + pi) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x < 0 and y!=0#
#x^2 + y^2 + sqrt(x^2 + y^2) = 2(tan^-1(y/x) + 2pi) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x > 0 and y<0#
Undefined elsewhere.
