Cross-Multiply the Equation

Multiply the denominator by the right-hand side constant 2 to eliminate the fraction.

$18 \cos \theta + 5 \sin \theta = 2(5 \cos \theta + 4 \sin \theta)$

Expand and Group Terms

Distribute the 2 across the brackets and collect all $\cos \theta$ terms on one side and $\sin \theta$ terms on the other.

$18 \cos \theta + 5 \sin \theta = 10 \cos \theta + 8 \sin \theta$

Simplify the Equation

Subtract $10 \cos \theta$ from both sides and $5 \sin \theta$ from both sides to isolate the trigonometric functions.

$18 \cos \theta - 10 \cos \theta = 8 \sin \theta - 5 \sin \theta \implies 8 \cos \theta = 3 \sin \theta$

Find cot theta

Divide both sides by $\sin \theta$ and by 8 to get the value of $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

$\frac{\cos \theta}{\sin \theta} = \frac{3}{8} \implies \cot \theta = \frac{3}{8}$