A) -1
B) 0
C) 1
D) None of these
Correct Answer: B
Solution :
\[{{\tan }^{2}}\theta =2{{\tan }^{2}}\varphi +1\Rightarrow 1+{{\tan }^{2}}\theta =2\,(1+{{\tan }^{2}}\varphi )\] Þ \[{{\sec }^{2}}\theta =2{{\sec }^{2}}\varphi \Rightarrow {{\cos }^{2}}\varphi =2{{\cos }^{2}}\theta \] Þ \[{{\cos }^{2}}\varphi =1+\cos 2\theta \Rightarrow {{\sin }^{2}}\varphi +\cos 2\theta =0\]. Trick: Let\[\theta ={{45}^{o}}\], then \[\varphi =0\] \[\therefore \ \cos (2\times {{45}^{o}})+{{\sin }^{2}}0=0+0=0\].
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