If`f(x)=sin^2x+sin^2(x+pi/3)+cosxcos(x+pi/3) and g(5/4)=1,` then `(gof)(x)` is ____________

Options

A. a polynomial of first degree in sin x and cos x
B. a constant function
C. a polynomial of second degree in sin x and cos x
D. none of these

Answer

B

Explanation

We have `f(x)=sin^(2)x+sin^(2)(x+(pi)/(3))+cos(x+(pi)/(3))cos x`
`f(x)=(1)/(2){1-cos 2x+1-cos(2x+(2pi)/(3))+cos(2x+(pi)/(3))+"cos"(pi)/(3)}`
`f(x)=(1)/(2)[(5)/(2)-{cos2x+cos(2x+(2pi)/(3)p)}+cos(2x+(pi)/(3))]`
`Rightarrow f(x)=(1)/(2)[(5)/(2)-2cos(2x+(pi)/(3))"cos" (pi)/(3)+cos(2x+(pi)/(3))]`
`f(x)=(5)/(4)"for all "x in R`
`therefore "g of "(x)=g (f(x))=g((5)/(4))=1"for all x"`
Hence, g of (x) is a constant functions.