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
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