Q. Prove that: 2tan50 + tan20 = cot20

Sometimes this question can be asked like this:
Prove that: 2tan50 + tan20 = tan70
For now, we have to prove: 
Prove that: 2tan50 + tan20 = cot20
We know, 
20 + 50 = 70
taking tan on both sides, we have 
tan(20 + 50) = tan70 
(tan 20 + tan 50) / (1 - tan 20.tan 50)  = tan70  [∵ tan(A+B)=(tan A+tan B)/(1-tanA.tanB)]

tan20 + tan50 = tan70  * (1 - tan20.tan50)

tan 20 + tan50  = tan70 - tan 20.tan 50.tan70          

tan20 + tan50  = tan70 - tan 20.tan50.cot 20       [∵ tan70 = tan (90-20) = cot 20]
tan20 + tan50  = tan70 - tan 20.tan50.(1/tan20)   [∵  cot20 = 1/tan20]
tan20 + tan 50  = tan70 - tan50
tan20 + tan50 +  tan50 = tan70 
2tan50 + tan20 = tan70
2tan50 + tan20 = cot20  [∵ tan70 = cot20]  Proved.