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.

\(\displaystyle{4}{{\sec}^{{2}}{x}}+{9}{{\csc}^{{2}}{x}}=\)

\(\displaystyle{\left({2}{\sec{{x}}}-{3}{\csc{{x}}}\right)}^{{2}}+{12}{\sec{{x}}}{\csc{{x}}}\)

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.