how to prove that (cot x) / (1+tan x) equal to (cot x -1) / (2 - sec^2 x)?

Question

sec^2 x as in sec power of 2 x..need to prove it ASAP.And do please provide me with the complete solution,pls..

Answer 1

Prove the identity.

(cot x) / (1 + tan x) = (cot x - 1) / (2 - sec² x)

Left Hand Side = (cot x) / (1 + tan x)

= (cot x)(1 - tan x) / [(1 + tan x)(1 - tan x)]

= (cot x - 1) / (1 - tan² x)

= (cot x - 1) / [1 - (sec² x - 1)]

= (cot x - 1) / (2 - sec² x) = Right Hand Side