If \(5\sin\theta + 12\cos\theta = 13\), where \(0 < \theta < \dfrac{\pi}{2}\), then what is \(\tan\theta + \cot\theta\) equal to?
\(\dfrac{169}{60}\)
Since \(5^{2}+12^{2}=13^{2}\), the maximum possible value of \(5\sin\theta+12\cos\theta\) is exactly 13, achieved only when \(\sin\theta=\dfrac{5}{13}\) and \(\cos\theta=\dfrac{12}{13}\). Then \(\tan\theta=\dfrac{5}{12}\), \(\cot\theta=\dfrac{12}{5}\), so \(\tan\theta+\cot\theta = \dfrac{5}{12}+\dfrac{12}{5} = \dfrac{25+144}{60} = \dfrac{169}{60}\).
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