The period of the function `f(x)=c^((sin^2x) +sin^2 (x+pi/3)+cosxcos(x+pi/3))` is (where `c` is constant)

Options

A. 1
B. `(pi)/(2)`
C. `pi`
D. None of these

Answer

D

Explanation

`sin^(2)x+sin^(2)(x+(pi)/(3))+cosxcos(x+(pi)/(3))`
`=sin^(2)x+((sinx)/(2)+(sqrt(3)cosx)/(2))^(2)+cosx((sinx)/(2)-(sqrt(3)cosx)/(2))`
`=sin^(2)x+(sin^(2)x)/(4)+(3cos^(2)x)/(4)+(cos^(2)x)/(2)`
`=(5sin^(2)x)/(4)+(5cos^(2)x)/(4)`
`=(5)/(4)`
Hence, `f(x)=c^(5//4)=` constant, which is periodic but has no fixed period.