1. **Stating the problem:** Simplify the expression $$\sin 2t + \sin 5t - \cot \theta - 23 + \frac{1}{2} \sin 2t + \cos \alpha$$ where $\cot$ is assumed to be $\cot \theta$ for some angle $\theta$. 2. **Combine like terms:** Notice that $\sin 2t$ appears twice, once as $\sin 2t$ and once as $\frac{1}{2} \sin 2t$. Add these together: $$\sin 2t + \frac{1}{2} \sin 2t = \left(1 + \frac{1}{2}\right) \sin 2t = \frac{3}{2} \sin 2t$$ 3. **Rewrite the expression:** Substitute the combined term back: $$\frac{3}{2} \sin 2t + \sin 5t - \cot \theta - 23 + \cos \alpha$$ 4. **Final simplified form:** The expression cannot be simplified further without additional information about the angles $t$, $\theta$, or $\alpha$. **Answer:** $$\boxed{\frac{3}{2} \sin 2t + \sin 5t - \cot \theta - 23 + \cos \alpha}$$