Find the least positive value of $\theta$.
$$\cot (6\theta +6^{\circ })=\frac{1}{\tan (7\theta +3^{\circ })}$$
$$\theta =$$

Step 1 ：The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, the equation can be simplified to: $\tan (6\theta +6^{\circ })=\tan (7\theta +3^{\circ })$

Step 2 ：The tangent function has a period of 180 degrees, so the two angles are equivalent if they differ by a multiple of 180 degrees. Therefore, we can write: $6\theta +6^{\circ }=7\theta +3^{\circ }+k\cdot {180}^{\circ }$ where k is an integer.

Step 3 ：Solving this equation for $\theta$ gives us $\theta =3-180k$. However, we want the least positive value of $\theta$, so we need to find the smallest positive integer value of k that makes $\theta$ positive.

Step 4 ：The solution to the equation is $\theta =3-180k$. However, we want the least positive value of $\theta$, so we need to find the smallest positive integer value of k that makes $\theta$ positive.

Step 5 ：The least positive value of $\theta$ is $\boxed{{\displaystyle 3^{\circ }}}$