## Trigonometry (11th Edition) Clone

$$\cos\beta(\sec\beta+\csc\beta)=1+\cot\beta$$
$$A=\cos\beta(\sec\beta+\csc\beta)$$ - 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. $$A=\frac{\sin\beta}{\sin\beta}+\frac{\cos\beta}{\sin\beta}$$ $$A=1+\cot\beta\hspace{1cm}\text{(Quotient Identity)}$$