Is this an identity, i.e. exactly true, or is it only an approximation?

It is exactly true, follow the link and see my answer to your previous question: the angles here are π/2 - θ₁, π/2 - θ₂, π/2 - θ₃, π/2 - θ₄ and π/2 - θ₅ (notations and corresponding Pythagorean triangles cited in my previous answer), so

(π/2 - θ₁) + (π/2 - θ₂) + (π/2 - θ₃) + (π/2 - θ₄) + (π/2 - θ₅) =

= 5π/2 - (θ₁ + θ₂ + θ₃ + θ₄ + θ₅) = 5π/2 - π/2 = 2π as required.

I notice that from some time both of us keep ourself amused very well - usually you ask interesting questions, I answer them. It is a pity that some distinguished contributors in Y!A Mathematical Forum have abandoned participation.

P.S. I have had a very interesting time with Y!A these years and intend to stay - answering mathematical questions is enjoyable game for me, this site offers many opportunities. Yet when I have come across to such questions I have usually seen answers from Gianlino, Scythian, late M.Daftary and many others - now I can not compete with them...