For the first question,
cos150/sin150=1/tan150
From R.H.S
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.
1. cos150/sin150=1/tan150
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
1.cos150=-sqrt3/2
2.sin150=1/2
3.tan150=-1/sqrt3
therefore: cos150/sin150=1/tan150
=-sqrt3/2over1/2 = 1/1/-sqrt3
=(-sqrt3/2)(2/1)=1*-sqrt3/1
=-sqrt3=-sqrt3
therefore: cos150/sin150=1/tan150