1. **Problem Statement:** Simplify the given trigonometric expressions and solve for $\theta$ where $\theta \leq 90^\circ$.\n\n1.1 Simplifications:\n\n1.1.1 Simplify $\sin 45^\circ + \sin 45^\circ$.\n\n- $\sin 45^\circ = \frac{\sqrt{2}}{2}$.\n- So, $\sin 45^\circ + \sin 45^\circ = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}$.\n\n1.1.2 Simplify $\cos(60^\circ + 30^\circ) - \sin^2 30^\circ + \sqrt{\tan 45^\circ}$.\n\n- $\cos(60^\circ + 30^\circ) = \cos 90^\circ = 0$.\n- $\sin 30^\circ = \frac{1}{2}$, so $\sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.\n- $\tan 45^\circ = 1$, so $\sqrt{\tan 45^\circ} = \sqrt{1} = 1$.\n- Combine: $0 - \frac{1}{4} + 1 = \frac{3}{4}$.\n\n1.2 Solve for $\theta$ given the conditions:\n\n1.2.1 Solve $\tan \theta = \sin 90^\circ$.\n\n- $\sin 90^\circ = 1$.\n- So, $\tan \theta = 1$.\n- $\tan \theta = 1$ at $\theta = 45^\circ$ within $0^\circ \leq \theta \leq 90^\circ$.\n\n1.2.2 Solve $\frac{1}{2 \sin \theta} = 1$.\n\n- Multiply both sides by $2 \sin \theta$: $1 = 2 \sin \theta$.\n- Divide both sides by 2: $\sin \theta = \frac{1}{2}$.\n- $\sin \theta = \frac{1}{2}$ at $\theta = 30^\circ$ within $0^\circ \leq \theta \leq 90^\circ$.\n\n**Final Answers:**\n- 1.1.1: $\sqrt{2}$.\n- 1.1.2: $\frac{3}{4}$.\n- 1.2.1: $\theta = 45^\circ$.\n- 1.2.2: $\theta = 30^\circ$.