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Question
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
Options
`sqrt29/3`
`29/3`
`sqrt3/29`
`3/29`
Solution
(d) `3/29`
\[\text{ Let }, \cos^{- 1} \frac{1}{5\sqrt{2}} = y \text{ and } \sin^{- 1} \frac{4}{\sqrt{17}} = z\]
\[\therefore \cos{y} = \frac{1}{5\sqrt{2}} \Rightarrow \sin{y} = \frac{7}{5\sqrt{2}} \Rightarrow \tan{y} = 7\]
\[\sin{z} = \frac{4}{\sqrt{17}} \Rightarrow \cos{z} = \frac{1}{\sqrt{17}} \Rightarrow \tan{z} = 4\]
\[\therefore \tan\left( \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right) = \tan\left( y - z \right)\]
\[ = \frac{\tan{y} - \tan{z}}{1 + \tan{y} \tan{z}}\]
\[ = \frac{7 - 4}{1 + 7 \times 4}\]
\[ = \frac{3}{29}\]
\[\therefore \tan\left( \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right) = \tan\left( y - z \right)\]
\[ = \frac{\tan{y} - \tan{z}}1 + \tan{y} \tan{z}\]
\[ = \frac{7 - 4}{1 + 7 \times 4}\]
\[ = \frac{3}{29}\]
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