Derivative function of inverse function

Calculus Level pending

Let \( g(x) \) be the derivative of the inverse function of \( f(x) = \tan^2x \) for \( 0 \lt x \lt \frac{\pi}{2} .\) If the minimum value of \( h(x)=\frac{-g'(x)}{\left(g(x)\right)^2} \) is \(a,\) which is attained at \(x=b,\) then what is \( a^{2} + 60b?\)

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