Question

Tan theta over 1 + sec theta + 1 +sec theta over Tan theta= 2 csc theta

Answer 1

Verify the identity:

Answer 2

totally confusing the way you typed.

Lucky that I know the relation is supposed to say:
tanθ/(1+secθ) + (1+secθ)/tanθ = 2cscθ

LS = (tan^2 θ + (1+secθ)^2)/(tanθ(1+secθ)) , using a common denominator
= (tan^2 θ + 1 + 2secθ + sec^2 θ)((tanθ(1+secθ))
= (sec^2 θ - 1 + 1 + 2secθ + sec^2 θ)/(tanθ(1+secθ))
= 2secθ(secθ + 1)/(tanθ(1+secθ))
= 2secθ/tanθ , secθ ≠ -1
= 2(1/cosθ)(cosθ/sinθ)
= 2/sinθ
= 2cscθ
= RS

Answer 3

extra credit: During Reiny's proof, he came to the step where he noted

= 2secθ/tanθ , secθ ≠ -1

Are there any other values you need to watch out for ?