What is the principal value of

By basic idea of ITF, we know that:

sin^-1(sin x) = x if x ∈ [-π/2, π/2]

and cos^-1(cos x) = x if x ∈ [0, π]

We know that range of cos^-1x is [0, π]

∵ 0<(2π/3)<π

∴ cos^-1(cos 2π/3) = 2π/3 …(1)

Whereas range of sin^-1x is [-π/2, π/2]

And 2π/3 ∉ [-π/2, π/2]

∴ we need to change it.

sin^-1(sin 2π/3) = sin^-1(sin (π-π/3)) = sin^‑1(sin π/3)

clearly, π/3 ∈ [-π/2, π/2]

∴ sin^‑1(sin π/3) = π/3 …(2)

Thus from 1 and 2 –

∴ cos^-1(cos 2π/3) + sin^-1(sin 2π/3) = 2π/3 + π/3 = π