1. **Simplify** $\frac{1 + \sec(-\theta)}{\sin(-\theta) + \tan(-\theta)}$. Recall the even-odd properties of trig functions: - $\sec(-\theta) = \sec \theta$ (secant is even) - $\sin(-\theta) = -\sin \theta$ (sine is odd) - $\tan(-\theta) = -\tan \theta$ (tangent is odd) Substitute these: $$\frac{1 + \sec \theta}{-\sin \theta - \tan \theta} = \frac{1 + \sec \theta}{-(\sin \theta + \tan \theta)} = - \frac{1 + \sec \theta}{\sin \theta + \tan \theta}$$ Rewrite $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$: $$- \frac{1 + \frac{1}{\cos \theta}}{\sin \theta + \frac{\sin \theta}{\cos \theta}} = - \frac{\frac{\cos \theta + 1}{\cos \theta}}{\sin \theta \left(1 + \frac{1}{\cos \theta}\right)}$$ Simplify denominator: $$\sin \theta \left(\frac{\cos \theta + 1}{\cos \theta}\right)$$ So expression is: $$- \frac{\frac{\cos \theta + 1}{\cos \theta}}{\sin \theta \frac{\cos \theta + 1}{\cos \theta}}$$ Cancel $\frac{\cos \theta + 1}{\cos \theta}$ top and bottom: $$- \frac{\cancel{\frac{\cos \theta + 1}{\cos \theta}}}{\sin \theta \cancel{\frac{\cos \theta + 1}{\cos \theta}}} = - \frac{1}{\sin \theta} = - \csc \theta$$ **Answer:** $- \csc \theta$ (option 5). --- 2. **Simplify** $\cot^2 y (\sec^2 y - 1)$. Recall identity: $$\sec^2 y - 1 = \tan^2 y$$ So expression becomes: $$\cot^2 y \cdot \tan^2 y$$ Recall $\cot y = \frac{1}{\tan y}$, so: $$\cot^2 y \tan^2 y = \left(\frac{1}{\tan y}\right)^2 \tan^2 y = 1$$ **Answer:** $1$ (option 3). --- 3. **Find** $\ln |\sec \theta|$. Recall $\sec \theta = \frac{1}{\cos \theta}$, so: $$\ln |\sec \theta| = \ln \left| \frac{1}{\cos \theta} \right| = - \ln |\cos \theta|$$ **Answer:** $- \ln |\cos \theta|$ (option 3). --- 4. **Simplify** $\cos^2 \beta - \sin^2 \beta$. Recall identity: $$\cos 2\beta = \cos^2 \beta - \sin^2 \beta$$ Also recall: $$\cos 2\beta = 2 \cos^2 \beta - 1$$ So: $$\cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta - 1$$ **Answer:** $2 \cos^2 \beta - 1$ (option 1).