A) \[-\frac{1}{\sqrt{2}}\]
B) 0
C) \[\frac{1}{\sqrt{2}}\]
D) \[\frac{1}{2\sqrt{2}}\]
Correct Answer: C
Solution :
[c] The given trigonometric expression is: \[\cos \left[ {{\tan }^{-1}}\left\{ \tan \left( \frac{15\pi }{4} \right) \right\} \right]\] \[=\cos \left[ {{\tan }^{-1}}\left\{ \tan \left( 4\pi -\frac{\pi }{4} \right) \right\} \right]\] \[=\cos \left[ {{\tan }^{-1}}\left\{ -\tan \frac{\pi }{4} \right\} \right]=\cos \left[ {{\tan }^{-1}}\tan \left( \frac{-\pi }{4} \right) \right]\] Since, \[{{\tan }^{-1}}\theta \] is defined for \[\frac{-\pi }{2}<\theta <\frac{-\pi }{2}\] \[=\cos \left( \frac{-\pi }{4} \right)\] \[\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}\] [since \[\cos (-\theta )=cos\theta \]]
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