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.