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If $2{\mathrm{tan}}^{2}x-5\mathrm{sec}x-1=0$ has 7 roots in $\left[0,\frac{n\pi }{2}\right]$ then the greatest value of n is?
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We have $\mathrm{cos}x=\frac{1}{3}.$
For n=1,5,9,13,⋯, the number of different roots in $\left[0,\frac{n\pi }{2}\right]$ is 1,3,5,7,⋯ respectively.
So, the greatest value of n is 13+2=15