Q. cosec40 - root(3)sec40 = 4cot100

Soln:
L.H.S.

= cosec40 - Ö3sec40 

= 1/sin40 - Ö3/cos40

= 1/sin40 - tan60/cos40  [tan60=Ö3]

= 1/sin40 - sin60/(cos40.cos60)

= (cos40.cos60 - sin40.sin60)/(sin40.cos40.cos60)

= cos(40+60)/(sin40.cos40.cos60)

= 2cos100/(2sin40.cos40.cos60)  [sin2.40=2sin40.cos40]

= 2cos100/[sin80.(1/2)]     [cos60=1/2]

= 4cos100/sin(180-80)

= 4cos100/sin100

= 4cot100 = R.H.S.