Expand the numerator:
(sin(x) + tan(x))² + cos²(x) - sec²(x)
= sin²(x) + 2 sin(x) tan(x) + tan²(x) + cos²(x) - sec²(x)
Recall the Pythagorean identity:
cos²(x) + sin²(x) = 1
Multiplying both sides by 1/cos²(x) gives an equivalent form,
1 + tan²(x) = sec²(x)
so that
tan²(x) - sec²(x) = -1
Then we have
(sin(x) + tan(x))² + cos²(x) - sec²(x)
= 1 + (-1) + 2 sin(x) tan(x)
= 2 sin(x) tan(x)
and so
((sin(x) + tan(x))² + cos²(x) - sec²(x))/tan(x)
= (2 sin(x) tan(x))/tan(x)
= 2 sin(x)
