The range of the following function is `f(x)=sqrt((1-cosx)sqrt((1-cosx)sqrt((1-cosx)sqrt(...oo))))` (a)(0,1) (b) `(0,1/2)` (c) `(0,2)` (d) `non eoft h e s e`

Options

A. [0, 1]
B. [0, 1/2]
C. [0, 2)
D. None of these

Answer

C

Explanation

Given `f(x)=sqrt((1-cosx)sqrt((1-cosx)sqrt((1-cosx)sqrt(...oo))))`
` =(1-cosx)^((1)/(2))(1-cosx)^((1)/(4))(1-cosx)^((1)/(8)) ...oo`
`=(1-cosx)^((1)/(2)+(1)/(4)+(1)/(8)+ ...oo)`
`=(1-cosx)^((1//2)/(1-(1//2)))`
`=1-cosx`
Thus, the range of `f(x)` is `[0,2).`