Which of the following are trigonometric identities, I have narrowed it down to two -

Question

(cscx + cotx)^2 = 1
OR
sin^2(x)sec^2(x) + 1 = cot^2(x)csc^2(x)

Answer 1

Before I go through the effort of proving a trig identity, I usually test it with some arbitrary value of an angle
Do not use one of the special angles, such as 30° or 45°, but pick something like 23°

for your first one:
LS = (csc 23° + cot 23°)^2
= (2.559 + 2.3558)^2
which is clearly not equal to 1

so the first one is not an identity.

Do the same with the second equation.