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Finding solutions to ${\displaystyle 3{\tan }^2\theta -\tan \theta -14=0}$ within ${\displaystyle 0\le \theta \le {360}^{\circ }}$

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How do you simplify
${\displaystyle {\tan }^{2}4b-{\sec }^{2}4b}$

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Find ${\displaystyle \frac{\sin x}{2},\frac{\cos x}{2}}$, and ${\displaystyle \frac{\tan x}{2}}$ from the given information.
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Minimizing ${\displaystyle 4{\sec }^2\left(x\right)+9{\csc }^2\left(x\right)}$ for x in the first quadrant. Discrepancy in solution
Using derivatives, I am able to show that the minimum value of the expression-in-question is equal to 25. I also verified this with Desmos graphing app. However, when I tried doing this using basic algebra, the answer turns out to be 26. I do not know why this is happening, but I should be extremely grateful to you if you can point out the error.
${\displaystyle 4{\sec }^{2}x+9{\csc }^{2}x=}$
${\displaystyle {(2\sec x-3\csc x)}^2+12\sec x\csc x}$
The value of the above expression will be minimum when that expression inside parenthesis equals zero; and that happens when ${\displaystyle \tan x=\frac{32}{}}$. Using this we can say that ${\displaystyle \sec x=\frac{\sqrt{13}}{2}}$ and ${\displaystyle \csc x=\frac{\sqrt{13}}{3}}$
If now we substitute these values in the above expression, answer turns out to be 26. I don't know why this is happening, but please help me find the error in this approach.

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