Question
If $$f\left( x \right) = {\sin ^2}x + {\sin ^2}\left( {x + \frac{\pi }{3}} \right) + \cos \,x\,\cos \left( {x + \frac{\pi }{3}} \right)$$ and $$g\left( {\frac{5}{4}} \right) = 1,$$ then $$gof\left( x \right) = ?$$
A.
$$1$$
B.
$$0$$
C.
$$\sin \,x$$
D.
none
Answer :
$$1$$
Solution :
$$\eqalign{
& {\text{We have,}} \cr
& f\left( x \right) = {\sin ^2}x + {\sin ^2}\left( {x + \frac{\pi }{3}} \right) + \cos \,x\,\cos \left( {x + \frac{\pi }{3}} \right) \cr
& = \frac{{1 - \cos \,2x}}{2} + \frac{{1 - \cos \left( {2x + \frac{{2\pi }}{3}} \right)}}{2} + \frac{1}{2}\left\{ {2\,\cos \,x\,\cos \left( {x + \frac{\pi }{3}} \right)\,} \right\} \cr
& = \frac{1}{2}\left[ {\frac{5}{2} - \left\{ {\cos \,2x + \cos \left( {2x + \frac{{2\pi }}{3}} \right)} \right\} + \cos \left( {2x + \frac{\pi }{3}} \right)} \right] \cr
& = \frac{1}{2}\left[ {\frac{5}{2} - 2\,\cos \left( {2x + \frac{\pi }{3}} \right)\cos \frac{\pi }{3} + \cos \left( {2x + \frac{\pi }{3}} \right)} \right] \cr
& = \frac{5}{4}{\text{ for all }}x \cr
& gof\left( x \right) = g\left( {f\left( x \right)} \right) = g\left( {\frac{5}{4}} \right) = 1\,\,\,\left[ {\because \,g\left( {\frac{5}{4}} \right) = 1\left( {{\text{given}}} \right)} \right] \cr
& {\text{Hence, }}gof\left( x \right) = 1,{\text{ for all }}x. \cr} $$