| If \[2\,\,({{\cos }^{2}}\theta -{{\sin }^{2}}\theta )=1,\]where \[\theta \] is a positive acute angle, then the value of \[\theta \] is [SSC (Assistant) 2012] |
A) \[60{}^\circ \]
B) \[30{}^\circ \]
C) \[45{}^\circ \]
D) \[22\frac{1}{2}{}^\circ \]
Correct Answer: B
Solution :
| \[2\,\,({{\cos }^{2}}\theta -{{\sin }^{2}}\theta )=1\] |
| \[\Rightarrow \]\[{{\cos }^{2}}\theta -{{\sin }^{2}}\theta =\frac{1}{2}\]\[\Rightarrow \]\[1-{{\sin }^{2}}\theta -{{\sin }^{2}}\theta =\frac{1}{2}\] |
| \[\Rightarrow \]\[1-2{{\sin }^{2}}\theta =\frac{1}{2}\]\[\Rightarrow \]\[2{{\sin }^{2}}\theta =1-\frac{1}{2}\] |
| \[\Rightarrow \]\[2{{\sin }^{2}}\theta =\frac{1}{2}\]\[\Rightarrow \]\[{{\sin }^{2}}\theta =\frac{1}{4}\] |
| \[\Rightarrow \]\[\sin \theta =\frac{1}{2}\]\[\Rightarrow \]\[\theta ={{\sin }^{-1}}\left( \frac{1}{2} \right)\] |
| \[\therefore \] \[\theta =30{}^\circ \] |
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