If $f (x) = \sin^{- 1} (x + 2019) - \cos^{- 1} (x + 2021)$ and $g (x) = \sin^{- 1} (x + 2017) + \cos^{- 1} (x + 2019)$ and $x_{1}, x_{2}$ lies in the domain of $f (x) a n d g (x)$ respectively, then the value of $| \lim_{x \to x_{1} + x_{2}} \frac{\sin (x - 2 x_{1} - 2) - \sin (2 x - 3 x_{1} + 2016)}{\tan (x - x_{1} - x_{2})} |$ is equal to

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Detailed Solution

$\begin{array}{l} f (x) = \sin^{- 1} (x + 2019) - \cos^{- 1} (x + 2021) \\ - 1 \leq x + 2019 \leq 1 - 1 \leq x + 2021 \leq 1 \\ - 2020 \leq x \leq - 2018 - 2022 \leq x \leq - 2020 \\ ∴ D o m a i n o f f (x) i s {- 2020} \\ g (x) = \sin^{- 1} (x + 2017) + \cos^{- 1} (x + 2019) \\ - 1 \leq x + 2017 \leq 1 - 1 \leq x + 2019 \leq 1 \\ - 2018 \leq x \leq - 2016 - 2020 \leq x \leq - 2018 \\ ∴ D o m a i n o f g (x) i s {- 2018} \\ x_{1} = - 2020 a n d x_{2} = - 2018 \\ | \lim_{x \to - 4038} \frac{\sin (x + 4038) - \sin (2 x + 8076)}{\tan (x + 4038)} | = 1 \end{array}$