Question

what is the difference between csc (1/sqrt 2) and sin^-1 (1/sqrt 2) and determine their values

Answer 1

csc(1/√2) is very unusual. It is 1.5393...

1/√2 is not an angle you see very often.

sin^-1(1/√2) = π/4
It is an angle whose sine is 1/√2

since csc(π/4) = √2,
csc^-1(√2) = π/4
In general,
sin^-1 x = csc^-1 1/x