Answer
$$\frac{\csc\theta}{\cot(-\theta)}=-\sec\theta$$
Work Step by Step
$$A=\frac{\csc\theta}{\cot(-\theta)}$$
- Negative-Angle Identity:
$$\cot(-\theta)=-\cot\theta$$
Replace into $A$:
$$A=\frac{\csc\theta}{-\cot\theta}$$
- Reciprocal Identity:
$$\csc\theta=\frac{1}{\sin\theta}$$
- Quotient Identity:
$$\cot\theta=\frac{\cos\theta}{\sin\theta}$$
Replace into $A$:
$$A=\frac{\frac{1}{\sin\theta}}{-\frac{\cos\theta}{\sin\theta}}$$
$$A=-\frac{1\times\sin\theta}{\sin\theta\cos\theta}$$
$$A=-\frac{1}{\cos\theta}$$
- Reciprocal Identity:
$$\sec\theta=\frac{1}{\cos\theta}$$
Therefore, $$A=-\sec\theta$$