The minimum value of `16 tan^2(theta) + 25 cot^2(theta)` is.
\[ \text { The minimum value of } 16 \tan ^2 \theta+25 \cot ^2 \theta \text { is is } \].
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The minimum value of `16 tan^2(theta) + 25 cot^2(theta)` is.
To find the minimum value of \(16\tan^2\theta + 25\cot^2\theta\), we can use the formula for the minimum value of the sum of two positive numbers \(a\) and \(b\), which is \(2\sqrt{ab}\) when \(a\) and \(b\) are positive. In this case, \(a = 16\) and \(b = 25\). So, the minimum value is \(2\sqrt{16 \times 25} = 2 \times 4 \times 5 = 40\).
Therefore, the minimum value of \(16\tan^2\theta + 25\cot^2\theta\) is \(40\).