If sin - 1 17 + cos - 1 4 5 - tan - 1 77 36 = 0 , 0 13 , then sin - 1 ( sin ) + cos - 1 ( cos ) is equal to [2023]

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Solution

Q.
If $\sin^{- 1} \frac{α}{17} + \cos^{- 1} \frac{4}{5} - \tan^{- 1} \frac{77}{36} = 0,$ $0 < α < 13,$ then $\sin^{- 1} (\sin α) + \cos^{- 1} (\cos α)$ is equal to [2023]

1 $16 - 5 π$

2 $16$

3 $0$

4 $π$

Ans.
(4)

We have,

$\sin^{- 1} \frac{α}{17} + \cos^{- 1} \frac{4}{5} - \tan^{- 1} \frac{77}{36} = 0; 0 < α < 13$

$\Rightarrow \sin^{- 1} \frac{α}{17} = \tan^{- 1} \frac{77}{36} - \cos^{- 1} \frac{4}{5} = \tan^{- 1} \frac{77}{36} - \sin^{- 1} \frac{3}{5}$

$\Rightarrow \frac{α}{17} = \sin (\tan^{- 1} \frac{77}{36} - \sin^{- 1} \frac{3}{5})$

$= \sin (\tan^{- 1} \frac{77}{36}) \cdot \cos (\sin^{- 1} \frac{3}{5}) - \cos (\tan^{- 1} \frac{77}{36}) \sin (\sin^{- 1} \frac{3}{5})$

$\Rightarrow \frac{α}{17} = \sin (\sin^{- 1} \frac{77}{85}) \cdot \cos (\cos^{- 1} \frac{4}{5}) - \cos (\cos^{- 1} \frac{36}{85}) (\frac{3}{5})$

$= \frac{77}{85} \times \frac{4}{5} - \frac{36}{85} \times \frac{3}{5} \Rightarrow α = 8$

$Now, \sin^{- 1} (\sin α) + \cos^{- 1} (\cos α) = \sin^{- 1} (\sin 8) + \cos^{- 1} (\cos 8)$

$= 3 π - 8 + 8 - 2 π [∵ 3 π - 8 \in [- \frac{π}{2}, \frac{π}{2}]] = π$

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