Prove that : (tanA+cotB)(cotA-tanB)=cotAcotB-tanAtanB

We'll manipulate the left side of trigonometric
identity.

We'll substitute cot B = 1/tan B and cot A =
1/tan A

We'll re-write the left
side:

(tanA+cotB)(cotA-tanB) = (tan A + 1/tan B)(1/tanA -
tan B)

We'll remove the brackets using FOIL
method:

(tan A + 1/tan B)(1/tanA - tan B) = tanA/tanA -
tanA*tanB + 1/tanA*tanB - tanB/tanB

We'll simplify, we'll
eliminate like terms and we'll get:

(tan A + 1/tan
B)(1/tanA - tan B) = 1/tanA*tanB - tanA*tanB

But
1/tanA*tanB = cotA*cotB

(tan A + 1/tan
B)(1/tanA - tan B) = cotA*cotB - tanA*tanB = RHS