1.

Find the period of the following function (i) `f(x) =|sinx|+|cosx|` (ii) `f(x)=cos(cosx)+cos(sinx)` (iii) `f(x)= (|sinx+cosx|)/(|sinx|+|cosx|)`

Answer» (i) `f(x) =|sinx|+|cosx|`
Period of both `|sinx|` and `|cosx|` is `pi`.
So, we can consider period of f(x) to be `pi`.
But `f((pi)/(2)+x)=|sin((pi)/(2)+x)|+|cos((pi)/(2)+x)|`
`=|cosx|+|-sinx|`
`=|cosx|+|sinx|`
`=f(x)`
So, period of f(x) is `Pi//2`.
(ii) `f(x)=cos(cosx)+cos(sinx)`
Period of `cos(cosx) " is " pi.`
Period of `cos(sinx) " is " pi.`
So, we can consider period of f(x) to be `pi`.
But `f((pi)/(2)+x)=cos(cos((pi)/(2)+x))+cos(sin((pi)/(2)+x))`
`=cos(-sinx)+cos(cosx)`
`=cos(sinx)+cos(cosx)`
So, period of f(x) is `pi//2`
(iii) `f(x)= (|sinx+cosx|)/(|sinx|+|cosx|)`
`|sinx +cosx|=sqrt(2)|sin(x+(pi)/(4))|`, which is has period `pi`.
Period of `|sinx|+|cosx| " is " pi//2`.
Therefore, period of f(x) is L.C.M. of `(pi,pi//2)=pi`


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