Consider the curves $C_{1}$ and $C_{2}$ Where $C_{1} : \sin^{- 1} x = \cos^{- 1} y$ $C_{2} : t a n^{- 1} x = \cot^{- 1} y$ and
$n_{1} =$ number of points of intersection of the curves $C_{1}$ and $C_{2}$
$n_{2} =$ number of tangents from $(- 10, - 8)$ to the curve $C_{2}$
$n_{3} =$ number of tangents from $(\frac{- 1}{2}, \frac{1}{2})$ to the curve $C_{2}$
$n_{4} =$ number of tangents from $(- 10, - 8)$ to the curve $C_{1}$
$n_{5} =$ number of tangents from $(\frac{- 1}{2024}, \frac{- 1}{2024})$ to the curve $C_{2}$ Then $n_{1} + n_{2} + n_{3} + n_{4} + n_{5} =$

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Detailed Solution

$S i n^{- 1} x = \cos^{- 1} y \Rightarrow x, y \in [0, 1] a n d x^{2} + y^{2} = 1 ∴ C_{1} : x^{2} + y^{2} = 1 \forall x, y \in [0, 1] \tan^{- 1} x = \cot^{- 1} y \Rightarrow x, y \in (0, \infty) a n d x y = 1 ∴ C_{2} : x y = 1, \forall x, y \in (0, \infty)$