A) 0
B) \[\frac{\pi }{4}\]
C) \[\frac{\pi }{6}\]
D) \[\frac{\pi }{2}\]
Correct Answer: A
Solution :
We have, \[{{\sin }^{-1}}\left[ \cos \left( {{\sin }^{-1}}\sqrt{\left( \frac{2-\sqrt{3}}{4} \right)}+{{\cos }^{-1}}\frac{\sqrt{12}}{4}+{{\sec }^{-1}}\sqrt{2} \right) \right]\] \[={{\sin }^{-1}}\left[ \cos \left( {{\sin }^{-1}}\left( \frac{\sqrt{3}-1}{2\sqrt{2}} \right)+{{\cos }^{-1}}\frac{\sqrt{3}}{2}+{{\cos }^{-1}}\frac{1}{\sqrt{2}} \right) \right]\] \[\left[ \because \,\,{{\left( \frac{2-\sqrt{3}}{4} \right)}^{1/2}}=\frac{\sqrt{3}-1}{2\sqrt{2}} \right]\] \[={{\sin }^{-1}}[\cos ({{15}^{o}}+{{30}^{o}}+{{45}^{o}})]\] \[={{\sin }^{-1}}[\cos {{90}^{o}}]\] \[\left( \because \,\,\sin {{15}^{o}}=\frac{\sqrt{3}-1}{2\sqrt{2}} \right)\] \[={{\sin }^{-1}}0\] \[=0\]
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