If \frac{\sin \theta+\cos \theta}{\sin \theta-\cos \theta}=5, where 0 \lt \theta \lt \frac{\pi}{2}, \theta \neq \frac{\pi}{4}, then what is the value of \frac{2 \sin \theta+3 \cos \theta}{3 \sin \theta-2 \cos \theta}?

Explanation

Dividing the given equation by \cos\theta, we get \frac{\tan\theta+1}{\tan\theta-1} = 5 \implies \tan\theta+1 = 5\tan\theta-5 \implies 4\tan\theta = 6 \implies \tan\theta = 1.5. Dividing the target expression by \cos\theta yields \frac{2\tan\theta+3}{3\tan\theta-2} = \frac{2(1.5)+3}{3(1.5)-2} = \frac{6}{2.5} = \frac{12}{5}.

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