Answer
$$\frac{\csc\theta\sec\theta}{\cot\theta}=\sec^2\theta$$
Work Step by Step
$$A=\frac{\csc\theta\sec\theta}{\cot\theta}$$
$$A=(\csc\theta\sec\theta)\times(\frac{1}{\cot\theta})$$
- Reciprocal Identities:
$$\csc\theta=\frac{1}{\sin\theta}\hspace{1cm}\sec\theta=\frac{1}{\cos\theta}$$
So, $$\csc\theta\sec\theta=\frac{1}{\sin\theta}\frac{1}{\cos\theta}=\frac{1}{\sin\theta\cos\theta}\hspace{1cm}(1)$$
- Another reciprocal identity:
$$\cot\theta=\frac{1}{\tan\theta}$$ which means $$\tan\theta=\frac{1}{\cot\theta}$$
Yet, according to a Quotient Identity: $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
Therefore, $$\frac{\sin\theta}{\cos\theta}=\frac{1}{\cot\theta}\hspace{1cm}(2)$$
Combine $(1)$ and $(2)$ back into $A$, we have
$$A=\frac{1}{\sin\theta\cos\theta}\times\frac{\sin\theta}{\cos\theta}$$
$$A=\frac{1}{\cos^2\theta}$$
$$A=\sec^2\theta\hspace{1.5cm}\text{(Reciprocal Identity)}$$
