If `f: R->R` be given by `f(x)=sin^2x+sin^2(x+pi//3)+cosx\ cos(x+pi//3)` for all `x in R` , and `g: R->R` be such that `g(5//4)=1` , then prove that `gof: R->R` is a constant function.

Explanation

`f (x) = sin^2x + sin^2 (x + (pi)/3) + cosxcos(x + (pi)/3)`


`= sin^2 x + [(1 / 2) sinx + (√3 / 2) cosx]^2 + cos x [(1 / 2) cosx – (√3 / 2) sinx]`


`= sin^2 x + (1 / 4) [√3 cosx + sinx]^2 + (1 / 2) [(cos^2 x – √3 sinx cosx]`


`= sin^2 x + (3 / 4) cos^2 x + (sin2 x) / 4 + (√3 / 2) sinx cosx + cos2 x / 2 + – (√3 / 2) sinx cosx`


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