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

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

open the brackets and multiply the terms:

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

use tan x * cot x = 1 and sec x * cos x = 1

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

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

**This proves that (tan A - sec B) (cot A + cos B) = tan A cos B - cot A sec B**