Prove the following: (tan A + cot B)(cot A - tan B) = cot A - tan A*tan B

I have replaced the theta with A and phi with B in the
identity to make it easier to type.

We have to prove: (tan
A + cot B)(cot A - tan B) = cot A - 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 = 1/cot x or
tan x * cot x = 1

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

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

The identity you have given has a missing term.
You have mistyped cot A instead of cot A*cot
B.

Therefore, the accurate
identity (tan A + cot B)(cot A - tan B) = cot A *cot B - tan A * tan B is
proved.