For the first question,
1/tan150 = ( 1/sin150/cos150 )
1/sin150/cos150 = 1 x cos150/sin150
= cos150/sin150 (proven)
For proving question try to have a formula sheet which contains all the trigo identities and ask yourself a few qns like what trigo identity can i use to solve the problem. You will be able to solve problem fairly easily with constant practice. Hope it helps.
sketch the angle in standard position. Then determine the values of x,y,r,using the 30 deg triangle and that it lies in the quadrant 2.
from the diagram, i can see that
=-sqrt3/2over1/2 = 1/1/-sqrt3