If 3\sin \theta + 4\cos \theta = 5, then what is a value of 4\tan \theta + 3\cot \theta ?

Explanation

Dividing by 5 gives \frac{3}{5}\sin\theta + \frac{4}{5}\cos\theta = 1. Comparing with \sin^2\theta + \cos^2\theta = 1, we get \sin\theta = \frac{3}{5} and \cos\theta = \frac{4}{5}. Thus \tan\theta = \frac{3}{4} and \cot\theta = \frac{4}{3}. The value is 4(\frac{3}{4}) + 3(\frac{4}{3}) = 3 + 4 = 7.

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