The main one is the Pythagorean identity,
sin²(x) + cos²(x) = 1
By dividing both sides by cos²(x), you get the tan-sec variant:
sin²(x)/cos²(x) + cos²(x)/cos²(x) = 1/cos²(x)
tan²(x) + 1 = sec²(x)
since tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x).
So the given expression reduces to
(sin²(x) + tan²(x) + cos²(x)) / sec²(x)
= (1 + tan²(x)) / sec²(x)
= sec²(x) / sec²(x)
= 1