We will be using the following:
cos(2t) = cos^2(t) - sin^2(t)
sin^2(t) + cos^2(t) = 1
sec(t) = 1/cos(t)
tan(t) = sin(t)/cos(t)
a^2 - b^2 = (a+b)(a-b)
a^3 + b^3 = (a+b)(a^2 -ab + b^2)
Starting from the right hand side:
(sec^2(t)-tan(t))/(sec(t)-tan(t)sec(t))= (1/cos^2(t)-sin(t)/cos(t))/(1/cos(t)-sin(t)/cos^2(t))
= (1/cos^2(t)-sin(t)/cos(t))/(1/cos(t)-sin(t)/cos^2(t)) * cos^2(t)/cos^2(t)
= (1 - sin(t)cos(t))/(cos(t) - sin(t))
= (sin^2(t) + cos^2(t) - sin(t)cos(t))/(cos(t) - sin(t))
= (sin^2(t) + cos^2(t) - sin(t)cos(t))/(cos(t) - sin(t)) * (sin(t)+cos(t))/(sin(t)+cos(t))
= ((cos(t)+sin(t))(sin^2(t) - sin(t)cos(t) + cos^2(t)))/((cos(t)+sin(t))(cos(t)-sin(t))
=(sin^3(t) + cos^3(t))/(cos^2(t)-sin^2(t))
= (sin^3(t) + cos^3(t))/cos(2t)