1. **Problem a:** Simplify $\cot x \sec x$. Recall the definitions: $$\cot x = \frac{\cos x}{\sin x}, \quad \sec x = \frac{1}{\cos x}$$ Multiply: $$\cot x \sec x = \frac{\cos x}{\sin x} \times \frac{1}{\cos x}$$ 2. Cancel common factors: $$= \frac{\cancel{\cos x}}{\sin x} \times \frac{1}{\cancel{\cos x}} = \frac{1}{\sin x}$$ 3. Recognize that $\frac{1}{\sin x} = \csc x$. **Answer a:** $\csc x$ --- 1. **Problem b:** Simplify $\frac{\sec^2 x - 1}{\sin^2 x}$. Recall the Pythagorean identity: $$\sec^2 x - 1 = \tan^2 x$$ Substitute: $$\frac{\sec^2 x - 1}{\sin^2 x} = \frac{\tan^2 x}{\sin^2 x}$$ 2. Express $\tan x$ as $\frac{\sin x}{\cos x}$: $$= \frac{\left(\frac{\sin x}{\cos x}\right)^2}{\sin^2 x} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\sin^2 x}$$ 3. Simplify the fraction: $$= \frac{\sin^2 x}{\cos^2 x} \times \frac{1}{\sin^2 x} = \frac{\cancel{\sin^2 x}}{\cos^2 x} \times \frac{1}{\cancel{\sin^2 x}} = \frac{1}{\cos^2 x}$$ 4. Recognize that $\frac{1}{\cos^2 x} = \sec^2 x$. **Answer b:** $\sec^2 x$ --- 1. **Problem c:** Simplify $$\frac{\cos^2 \left( \frac{\pi}{2} - x \right)}{\cos x}$$ Recall the co-function identity: $$\cos \left( \frac{\pi}{2} - x \right) = \sin x$$ 2. Substitute: $$= \frac{\sin^2 x}{\cos x}$$ **Answer c:** $\frac{\sin^2 x}{\cos x}$ --- 1. **Simplify:** $\sin \phi (\csc \phi - \sin \phi)$ Recall: $$\csc \phi = \frac{1}{\sin \phi}$$ 2. Substitute: $$= \sin \phi \left( \frac{1}{\sin \phi} - \sin \phi \right) = \sin \phi \left( \frac{1 - \sin^2 \phi}{\sin \phi} \right)$$ 3. Cancel $\sin \phi$: $$= \cancel{\sin \phi} \times \frac{1 - \sin^2 \phi}{\cancel{\sin \phi}} = 1 - \sin^2 \phi$$ 4. Use Pythagorean identity: $$1 - \sin^2 \phi = \cos^2 \phi$$ **Answer:** $\cos^2 \phi$ --- 1. **Simplify:** $\cot \left( \frac{\pi}{2} - x \right) \cos x$ Recall co-function identity: $$\cot \left( \frac{\pi}{2} - x \right) = \tan x$$ 2. Substitute: $$= \tan x \cos x$$ 3. Express $\tan x$ as $\frac{\sin x}{\cos x}$: $$= \frac{\sin x}{\cos x} \times \cos x$$ 4. Cancel $\cos x$: $$= \cancel{\cos x} \times \frac{\sin x}{\cancel{\cos x}} = \sin x$$ **Answer:** $\sin x$ --- 1. **Simplify:** $(\cos t)(1 + \tan^2 t)$ Recall Pythagorean identity: $$1 + \tan^2 t = \sec^2 t$$ 2. Substitute: $$= \cos t \times \sec^2 t = \cos t \times \frac{1}{\cos^2 t} = \frac{\cos t}{\cos^2 t}$$ 3. Simplify: $$= \frac{\cancel{\cos t}}{\cos^2 t} = \frac{1}{\cos t} = \sec t$$ **Answer:** $\sec t$ --- 1. **Simplify:** $\sec^2 x \tan^2 x + \sec^2 x$ 2. Factor out $\sec^2 x$: $$= \sec^2 x (\tan^2 x + 1)$$ 3. Use Pythagorean identity: $$\tan^2 x + 1 = \sec^2 x$$ 4. Substitute: $$= \sec^2 x \times \sec^2 x = \sec^4 x$$ **Answer:** $\sec^4 x$