| Value of \[{{\sec }^{2}}\theta +\frac{{{\sin }^{2}}\theta -2{{\sin }^{4}}\theta }{2{{\cos }^{4}}\theta -{{\cos }^{2}}\theta }\]is [SSC (CGL) Mains 2014] |
A) \[1\]
B) \[2\]
C) \[-\,\,1\]
D) \[0\]
Correct Answer: A
Solution :
| \[{{\sec }^{2}}\theta -\frac{{{\sin }^{2}}\theta -2{{\sin }^{4}}\theta }{2{{\cos }^{4}}\theta -{{\cos }^{2}}\theta }\] |
| \[={{\sec }^{2}}\theta =\frac{{{\sin }^{2}}\theta \,(1-2{{\sin }^{2}}\theta )}{{{\cos }^{2}}\theta \,(2{{\cos }^{2}}\theta -1)}\] |
| \[={{\sec }^{2}}\theta -\frac{{{\sin }^{2}}\theta \cdot \cos 2\theta }{{{\cos }^{2}}\theta \cdot \cos 2\theta }\] |
| \[[\because \cos 2\theta =1-2{{\sin }^{2}}\theta =2{{\cos }^{2}}\theta -1]\] |
| \[={{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1\] |
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