We have to prove that: (tan A + cot B) (cot A - tan B) = cot A cot B - tan A tan B

(tan A + cot B) (cot A - tan B)

open the brackets and multiply the terms.

=> tan A * cot A - tan A * tan B + cot A * cot B - cot B * tan B

Use the relation tan x * cot x = 1

=> 1 - tan A * tan B + cot A * cot B - 1

=> - tan A * tan B + cot A * cot B

=> cot A * cot B - tan A * tan B

**This proves the identity (tan A + cot B) (cot A - tan B) = cot A cot B - tan A tan B**