The range `f(x)=cos2x-sin2x` contains the set

Options

A. [2, 4]
B. `[-1, 1]`
C. `[-sqrt(2), sqrt(2)]`
D. `(-sqrt(2),2)`

Answer

C

Explanation

Let f(x)=cos2x-sin2x
`f(x)=(1)/(sqrt(2))[sqrt(2)cos2x-sin2x]`
`f(x)=sqrt(2)[(1)/(sqrt(2))cos2x-(1)/(sqrt(2))sin2x]`
`f(x)=sqrt(2)[cos.(pi)/(4)cos2x-sin.(pi)/(4)sin2x]`
`f(x)=sqrt(2)[cos((pi)/(4)+2x)]`
We know, `-1lecos((pi)/(4)+2x)le1`
`implies-sqrt(2)lesqrt(2)cos((pi)/(4)+2x)lesqrt(2)`
`implies-sqrt(2)lef(x)lesqrt(2)`
`therefore " Range of "f(x)=[-sqrt(2),sqrt(2)]`