1. The problem is to find the range of the variable $\theta$ given the inequality $0 < \theta \leq 360$ and the expression $\cos^2 x + \sin^2 x$. 2. Recall the Pythagorean identity in trigonometry: $$\cos^2 x + \sin^2 x = 1$$ This identity holds for all real values of $x$. 3. Since the expression $\cos^2 x + \sin^2 x$ is always equal to 1, it does not depend on $\theta$ or $x$. 4. Therefore, for any $\theta$ in the range $0 < \theta \leq 360$, the value of $\cos^2 x + \sin^2 x$ is always 1. 5. In conclusion, the expression is constant and equal to 1 for the given range of $\theta$.