Trigonometric Identities Question 288
Question: If $ \cos \theta =\frac{3}{5} $ and $ \cos \varphi =\frac{4}{5}, $ where $ \theta $ and $ \varphi $ are positive acute angles, then $ \cos \frac{\theta -\varphi }{2}= $
[MP PET 1988]
Options:
A) $ \frac{7}{\sqrt{2}} $
B) $ \frac{7}{5\sqrt{2}} $
C) $ \frac{7}{\sqrt{5}} $
D) $ \frac{7}{2\sqrt{5}} $
Show Answer
Answer:
Correct Answer: B
Solution:
We have $ \cos \theta =\frac{3}{5} $ and $ \cos \varphi =\frac{4}{5} $ . Therefore $ \cos (\theta -\varphi )=\cos \theta \cos \varphi +\sin \theta \sin \varphi $ $ =\frac{3}{5}.\frac{4}{5}+\frac{4}{5}.\frac{3}{5}=\frac{24}{25} $ But $ 2{{\cos }^{2}}( \frac{\theta -\varphi }{2} )=1+\cos (\theta -\varphi )=1+\frac{24}{25}=\frac{49}{50} $ \ $ {{\cos }^{2}}( \frac{\theta -\varphi }{2} )=\frac{49}{50} $ . Hence, $ \cos ( \frac{\theta -\varphi }{2} )=\frac{7}{5\sqrt{2}} $ .
BETA
