Prove the following identitie PLEASE SHOW YOUR WORK IN DETAI

Prove the following identitie.

PLEASE SHOW YOUR WORK IN DETAILS.

Prove the following identities. 2/1 + cosx - Tan^2 (x/2) = 1

Solution

we know that 1 + cos(2x) = 2cos²(x)

or 1 + cos(x) = 2cos²(x/2)

=> 1/(1 + cos(x)) = 1/2cos²(x/2) = (1/2)sec²(x/2)

=> 2/(1 + cos(x)) = sec²(x/2)

=> 2/(1 + cos(x)) - tan²(x/2) =  sec²(x/2) - tan²(x/2)

= 1 (using identity sec²(x) = 1 + tan²(x) )