trignometry

Consider  cot 30+cot 40 = cos(30)/sin(30) + cos(40)/sin(40)

= [sin(40)cos(30) + sin(30)cos(40)] / [sin(30)sin(40)]

Using sin (A+B)=sin Acos B+cos A sin B
= sin(70) / [(1/2)sin(40)] = cos(90 - 70) / [(1/2)sin(40)]

= cos(20) / [sin(20)cos(20)]  [using sin 2 A=2sin A cos A]

= 1/sin(20) = cosec(20)

So we have:

cot(30) + cot(40) = cosec(20)

Rearranging  the terms

cosec20-cot40 =cot(30)= √3