The period of `cosx^2` is

Options

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

Answer

D

Explanation

Let `f(x)= cosx^(2)` be a periodic function with period I, Then,
`f(x+T)=f(x)` for all `x in R `
`implies cos(x+T)^(2)=cos^(2)`, for all ` x in R `
`implies (x+T)^(2)=2n pi pm x^(2)`, for all ` x in R `
`implies x^(2)+2Tx+T^(2)pm x^(2)=2npi`, for all ` x in R `
This relation is not possible , because the right hand side is an integral multiple of 2 pi whereas the left hand side is a quadratic function of x.
For example , for x=0 and x=T, we obtain
`T^(2)=2npi and 4t^(2) pm T^(2)=2 n pi`
`implies 4T^(2) pm T^(2)=T^(2)`
`implies 4 pm=1" " [ :' (T gt 0), " which is not true "]`
Hence , `f(x)= cos^(2)x` is not a periodic functions.