1. Given that $\sin a = \cos b$ and $a, b$ are acute angles, recall that $\sin a = \cos (90^\circ - a)$, so $b = 90^\circ - a$. We want to find $\tan (a+b)$. 2. Substitute $b = 90^\circ - a$: $$\tan(a+b) = \tan(a + 90^\circ - a) = \tan 90^\circ$$ 3. $\tan 90^\circ$ is undefined, so $\tan(a+b)$ is undefined. 4. Next, if $\sin a = \cos b$ and $a,b$ acute, find $\tan(a+b)$. From step 1, $b = 90^\circ - a$, so $\tan(a+b) = \tan 90^\circ$ undefined as before. 5. If $\tan 5x = \cot 4x$, then $\tan 5x = \tan(90^\circ - 4x)$, so $5x = 90^\circ - 4x$ which implies $9x = 90^\circ$ and $x = 10^\circ$. Then $\sin 3x = \sin 30^\circ = \frac{1}{2}$. 6. Given $\sin 2\theta \sec \theta = 1$ with $\theta \in [0^\circ, 90^\circ]$. Rewrite using $\sec \theta = \frac{1}{\cos \theta}$: $$\sin 2\theta \cdot \frac{1}{\cos \theta} = 1$$ $$\frac{2 \sin \theta \cos \theta}{\cos \theta} = 1$$ $$2 \sin \theta = 1$$ $$\sin \theta = \frac{1}{2}$$ so $\theta = 30^\circ$. 7. If $5 \cos (270^\circ + \theta) = 4$ and $0 < \theta < 90^\circ$, use $\cos (270^\circ + \theta) = -\sin \theta$: $$5(-\sin \theta) = 4 \implies -5 \sin \theta = 4 \implies \sin \theta = -\frac{4}{5}$$ impossible for $\theta$ acute. Recheck: $\cos(270^\circ + \theta) = \sin \theta$? Actually, $\cos(270^\circ + \theta) = \sin \theta$? No, $\cos(270^\circ + \theta) = -\sin \theta$. Since $0 < \theta < 90^\circ$, $\sin \theta > 0$, so $\sin \theta = -\frac{4}{5}$ can't be. Possibly $\cos(270^\circ + \theta) = -\sin \theta$ is correct, so equation is unsolvable under given conditions unless $\theta$ is reinterpreted. Otherwise, no solution in $0<\theta<90^\circ$. 8. $\sin(-30^\circ) = -\sin 30^\circ = -\frac{1}{2}$. 9. $\cos 240^\circ + \cos 420^\circ$, reduce angles: $\cos 240^\circ = -\frac{1}{2}$, and $\cos 420^\circ=\cos 60^\circ = \frac{1}{2}$, sum is 0. 10. If $2 \sin x = -1$ and $x$ is acute, $\sin x = -\frac{1}{2}$ impossible since sine is positive in $(0, 90^\circ)$. No solution. 11. The range of $f(x) = \sin x$ is $[-1, 1]$. 12. Extra exercises involve angle co-functions and periodic identities using trigonometric properties and unit circle definitions. Final answers summary: 1) $\tan(a+b)$ is undefined. 3) $\sin 3x = \frac{1}{2}$. 6) $\theta = 30^\circ$. 8) No solution for $x$ positive acute with $2 \sin x = -1$. 9) Range of $f(x)=\sin x$ is $[-1, 1]$.