Write the principal value of tan(1) + cos(-1/2)

Consider tan(1)
Let y = tan(1)
Since 1 is positive, principal value of θ is π/4 ⇒ tan y = tan π/4
Range of principal value of tan is (-π/2, π/2)
Hence, the principal value of tan(1) is π/4
Consider cos(-1/2)
Let y = cos(-1/2) ⇒ cos y = -1/2
Since -1/2 is negative, principal value is π - θ i.e., π - π/3 = 3π - π/3 = 2π/3 ⇒ cos y = cos 2π/3
Range of principal value of cos is [0, π]
Hence, the principal value is 2π/3
Thus, we have tan(1) = π/4 and cos(-1/2) = 2π/3
∴ tan(1) + cos(-1/2) = π/4 + 2π/3 = 3π + 8π/12 = 11π/12