1.

For 0ltxlt(pi)/(6), all the values of tan^(2)3x cos^(2)x-4tan3xsin2x+16sin^(2)x lie in the interval

Answer»

`(0,(121)/(36))`
`(1,(121)/(9))`
`(-1,0)`
none of these

Solution :We have,
`tan^(2)3x cos^(2)x-4tan3xdsin2x+16sin^2)x`
`=((tan^(2)3x)/(tan^(2)x)-(8tan3x)/(tanx))sin^(2)x=((tan3x)/(tanx)-)^(2)sin^(2)x`
We KNOWN that
`1/3lt(tan3x)/(tanx)lt3`
`implies-11/3lt(tan3x)/(tanx)-4lt-1implies1lt((tan3x)/(tanx)-4)^(2)lt(121)/(9)`
Also, `0ltxlt(pi)/(9)implies0sinxlt1/2implies0ltsin^(2)xlt1/4`
`therefore0lt((tan3x)/(tanx)-4)^(2)sin^(2)xlt(121)/(36)`


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