We have to prove that :((sin x + cos x)/(sin x - cos x)) + ((sin x - cosx)/(sin x + cos x)) = (2*sec^2x/tan^2x) - 1

((sin x + cos x)/(sin x - cos x)) + ((sin x - cosx)/(sin x + cos x))

=> ((sin x + cos x)^2 + (sin x - cos x)^2)/(sin x + cos x)(sin x - cos x)

=> ((sin x)^2 + (cos x)^2 + 2*(sin x)(cos x) + (sin x)^2 + (cos x)^2 - 2*(cos x)(sin x))/(sin x + cos x)(sin x - cos x)

=> 2/[(sin x)^2 - (cos x)^2] ...(1)

(2*sec^2x/tan^2x) - 1

=> (2*sec^2x - tan^2x)/tan^2x

=> [(2/(cos x)^2 - (sin x)^2/(cos x)^2)]/[(sin x)^2/(cos x)^2]

=> [2 - (sin x)^2]/(sin x)^2 ...(2)

As we can see (1) is not equal to (2)

**The given relation is not an identity.**