2008/002) Prove the identity.

Prove the identity.

csc(20°) - cot(40°) = tan(60°)

Proof:
LHS= 1/sin (20°) – cos (40°) /sin (40°)
= 1/sin (20°) - cos(40°) /(2 sin (20°) cos (20°)
= ( 2 cos (20°) – cos (40°) )/ (2 sin (20°) cos (20°))
= ( 2 cos (20°) – cos (40°) )/ ( sin (40°))

Now numerator
= 2 cos (20°) – cos (40°)
= 2 cos (30°-10°) – (cos (30°+10°)
= 2 (cos (30°) cos (10°)+ sin (30°) sin (10°)) – (cos (30°) cos (10°)- sin (30°) sin (10°))
= cos (30°) cos (10°)+ 3 sin (30°) sin (10°)
= sqrt(3)/2 cos (10°) + 3/2 sin (10°)
= sqrt(3)( cos (10°)/2 + sqrt(3)/2 sin (10°) (take sqrt(3) common as it is tan 60)
= sqrt(3)( cos (10°) sin (30°) + sin (30°) sin (10°))
= sqrt(3)(sin (30°+10°))
= sqrt(3)(sin (40°)

so LHS = ( 2 cos (20°) – cos (40°) )/ sin (40°)
= sqrt(3)(sin (40°)/sin (40°)
= sqrt(3)