prove the following: (tan A - sec B) (cot A + cos B) = tan A cos B - cot A sec B

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