Evaluate Trigonometric Expression

The problem asks for the value of the expression $\frac{\cos \theta - \tan \theta}{2 \cot \theta}$ given that $\sin \theta = \frac{5}{13}$ and $\theta$ is a positive acute angle.

Calculate Trigonometric Values

First, find $\cos \theta$. Using the identity $\sin^2 \theta + \cos^2 \theta = 1$, and knowing $\theta$ is acute (so $\cos \theta > 0$):

$ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} $

$ \cos \theta = \sqrt{\frac{144}{169}} = \frac{12}{13} $

Next, find $\tan \theta$ and $\cot \theta$:

$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/13}{12/13} = \frac{5}{12} $

$ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{5/12} = \frac{12}{5} $

Evaluate the Expression

Substitute the calculated values into the given expression:

$ \frac{\cos \theta - \tan \theta}{2 \cot \theta} = \frac{\frac{12}{13} - \frac{5}{12}}{2 \times \frac{12}{5}} $

Calculate the numerator:

$ \frac{12}{13} - \frac{5}{12} = \frac{12 \times 12 - 5 \times 13}{13 \times 12} = \frac{144 - 65}{156} = \frac{79}{156} $

Calculate the denominator:

$ 2 \times \frac{12}{5} = \frac{24}{5} $

Divide the numerator by the denominator:

$ \frac{79/156}{24/5} = \frac{79}{156} \times \frac{5}{24} = \frac{79 \times 5}{156 \times 24} = \frac{395}{3744} $

The value of the expression is $\frac{395}{3744}$.