for question 2)

cos(sinx) >= 0

implies    -pi/2 <= sinx <= pi/2  (as cos is positive in first & fourth quadrant)

=>      -1.57 <= sinx  <= 1.57 --------------------------1

but we know that    -1 <= sinx <= 1 for x belonging to R -------------------2

from 1 & 2 domain is R