Trigonometric Identities Question 334
Question: If $ {{\tan }^{2}}\theta =2{{\tan }^{2}}\varphi +1, $ then $ \cos 2\theta +{{\sin }^{2}}\varphi $ equals
Options:
A) -1
B) 0
C) 1
D) None of these
Show Answer
Answer:
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 $ .
BETA
