_ = 1 ^ 10 ^ - 1 ( ( - 6 ) ) is equal to- - Vidyadip

Question

$\sum\limits_{\lambda = 1}^{10} {{{\sin }^{ - 1}}\left( {\sin \left( {\lambda \pi - \frac{\pi }{6}} \right)} \right)} $ is equal to-

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Answer

c
$ = \sum\limits_{n = 1}^5 {{{\sin }^{ - 1}}\left( {\sin \left( {2n\pi - \frac{\pi }{6}} \right)} \right) + } \sum\limits_{n = 1}^5 {{{\sin }^{ - 1}}\left( {\sin \left( {2n - 1} \right)\pi - \frac{\pi }{6}} \right)} $

$ = \sum\limits_{n = 1}^5 {{{\sin }^{ - 1}}\left( { - \sin \frac{\pi }{6}} \right) + \sum\limits_{n = 1}^5 {{{\sin }^{ - 1}}\left( {\sin \frac{\pi }{6}} \right)} } $

$ = - \frac{{5\pi }}{6} + \frac{{5\pi }}{6} = 0$

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