Answer
$$\cos\beta(\sec\beta+\csc\beta)=1+\cot\beta$$
Work Step by Step
$$A=\cos\beta(\sec\beta+\csc\beta)$$
- Reciprocal Identities:
$$\sec\beta=\frac{1}{\cos\beta}\hspace{2cm}\csc\beta=\frac{1}{\sin\beta}$$
So, $$\sec\beta+\csc\beta=\frac{1}{\cos\beta}+\frac{1}{\sin\beta}=\frac{\sin\beta+\cos\beta}{\sin\beta\cos\beta}$$
Therefore, $$A=\cos\beta\times\frac{\sin\beta+\cos\beta}{\sin\beta\cos\beta}$$
$$A=\frac{\sin\beta+\cos\beta}{\sin\beta}$$
Now we need to separate the numerator to eliminate the quotient.
$$A=\frac{\sin\beta}{\sin\beta}+\frac{\cos\beta}{\sin\beta}$$
$$A=1+\cot\beta\hspace{1cm}\text{(Quotient Identity)}$$
