If `T` is the period of the function `f(x)=[8x+7]+|tan2pix+cot2pix|-8x]` (where [.] denotes the greatest integer function), then the value of `1/T` is ___________

Answer

0.25

Explanation

`f(x)=[8x+7]+|tan2pi x+cot 2pi x|-8x`
`=[8x]-8x-7+|tan 2 pi x+cot2pi x|`
`= -{8x}+|tan2pi x+cot 2pi x|+7`
Period of {8x} is 1/8.
Also, `|tan2pi x+cot 2pi x|=|(sin2pi x)/(cos2pi x)+(cos2pi x)/(sin2pi x)|`
`=|(1)/(sin2pi x cos2pi x)|=|2"cosec"4pi x|`
Now, period of ` 2 "cosec" 4pi x` is 1/2. Then period of `|2"cosec"4pi x|` is 1/4.
Therefore, period is L.C.M of `(1)/(8) and (1)/(4)` which is `(1)/(4)`.