tan−1⁡(tan⁡1−θ)=1−θ, when

Clearly, 1-0 is real, if θ≤1.tan−1⁡(tan⁡1−θ)=1−θ⇒ 0≤1−θ<π2and θ≤1⇒ 0≤1−θ<π24and θ≤1⇒ −1≤−θ≤π24−1and θ≤1⇒ 1−π24<θ≤1 and θ≤1⇒4−π24<θ≤1