What is the <strong>MINIMUM</strong> value of \cos^{3}\theta+\sec^{3}\theta where 0^{\circ}\leq\theta\lt90^{\circ}?

Explanation

Since \sec\theta = \frac{1}{\cos\theta} and \cos\theta \gt 0 in the given range, we can apply the AM-GM inequality: x + \frac{1}{x} \geq 2. Therefore, \cos^3\theta + \sec^3\theta \geq 2, with the minimum value occurring at \theta = 0^{\circ}.

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