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Prove that the values of the function (s...

Prove that the values of the function `(sinxcos3x)/(sin3xcosx)` cannot lie between `1/3` and `3` for any real `x`

Text Solution

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Let `y=(sinxcos3x)/(sin3x cos x)=(tanx)/(tan 3x)`
`implies y = (tanx)/(tan3x)=(tanx(1-3tan^(2)x))/(3tanx-tan^(3)x)`
`=(1-3tan^(2)x)/(3-tan^(2)x)" "[because x ne0]`
Put tan x = t
`implies y = (1-3t^(2))/(3-t^(2))`
`implies 3y-t^(2)y = 1-3t^(2)`
`implies 3y-1=t^(2)y-3t^(2)`
`implies 3y-1=t^(2)(y-3)`

`implies (3y-1)/(y-3)=t^(2)implies(3y-1)/(y-3)gt0`
`therefore t^(2) gt 0`
NOTE It is a brilliant technique to convert equation into inequation and asked in IIT papers frequently. `impliesylt1//3orygt3`. This shows that y cannot lie between `1//3` and 3.
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