tan−1⁡(tan⁡1−θ)=1−θ, when - Infinity Learn by Sri Chaitanya

Clearly, $\sqrt{1 - 0}$ is real, if $θ \leq 1$ .

$\tan^{- 1} (\tan \sqrt{1 - θ}) = \sqrt{1 - θ}$

$\Rightarrow 0 \leq \sqrt{1 - θ} < \frac{π}{2}$ and $θ \leq 1$

$\Rightarrow 0 \leq 1 - θ < \frac{π^{2}}{4}$ and $θ \leq 1$

$\Rightarrow - 1 \leq - θ \leq \frac{π^{2}}{4} - 1$ and $θ \leq 1$

$\Rightarrow 1 - \frac{π^{2}}{4} < θ \leq 1$ and $θ \leq 1 \Rightarrow \frac{4 - π^{2}}{4} < θ \leq 1$

$\tan^{- 1} (\tan \sqrt{1 - θ}) = \sqrt{1 - θ}$ , when