45. \( \sin \theta \cot \theta=\cos \theta \)

To solve the equation \( \sin \theta \cot \theta=\cos \theta \), we can start by expressing \( \cot \theta \) as \( \frac{\cos \theta}{\sin \theta} \). This leads us to rewrite the original equation: \[ \sin \theta \left( \frac{\cos \theta}{\sin \theta} \right) = \cos \theta \] This simplifies to: \[ \cos \theta = \cos \theta \] This identity holds true for all angles \( \theta \) where \( \cos \theta \neq 0 \), so we must identify the values for \( \theta \) specifically. The condition \( \cos \theta = 0 \) occurs at \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \), where our original equation becomes undefined. Therefore, the solutions to the equation are all angles of the form \( \theta \neq \frac{\pi}{2} + n\pi \) or any angle for which \( \cos \theta \) is not equal to zero. In conclusion, the equation holds for all angles except those where \( \cos \theta = 0 \). So, to summarize, the solutions are: \[ \theta \in \mathbb{R} \quad \text{excluding} \quad \theta = \frac{\pi}{2} + n\pi, \text{ where } n \in \mathbb{Z}. \]