Functions Question 741
Question: If $ f(x)=sin^{2}x+{{\sin }^{2}}( x+\frac{\pi }{3} )+\cos x\cos ( x+\frac{\pi }{3} ) $ and $ g( \frac{5}{4} )=1, $ then gof(x)=
Options:
A) 1
B) 0
C) $ \sin x $
D) None
Show Answer
Answer:
Correct Answer: A
Solution:
[a] We have $ f(x)=sin^{2}x+{{\sin }^{2}}(x+\pi /3)+\cos x\cos (x+\pi /3) $ $ =\frac{1-\cos 2x}{2}+\frac{1-\cos (2x+2\pi /3)}{2}+ $ $ \frac{1}{2}{2cosx,cos(x+\pi /3)} $ $ =\frac{1}{2}[ \frac{5}{2}-{ \cos 2x+\cos ( 2x+\frac{2\pi }{3} ) }+\cos ( 2x+\frac{\pi }{3} ) ] $ $ =\frac{1}{2}[ \frac{5}{2}-2\cos ( 2x+\frac{\pi }{3} )\cos \frac{\pi }{3}+\cos ( 2x+\frac{\pi }{3} ) ] $ $ =\frac{5}{4} $ For all x. $ gof(x)=g(f(x))=g( \frac{5}{4} )=1 $ $ [\because g( \frac{5}{4} )=1 $ (Given)] Hence, $ gof(x)=1, $ for all x.
BETA
