# How can I simplify this expression? Sin(ß)Cos(-ß)Csc(ß)

Mar 25, 2018

It simplifies to $\cos \beta$.

#### Explanation:

Use the reciprocal definition, then the cosine difference formula:

$\textcolor{w h i t e}{=} \sin \beta \cos \left(- \beta\right) \csc \beta$

$= \sin \beta \cos \left(- \beta\right) \cdot \frac{1}{\sin} \beta$

$= \textcolor{red}{\cancel{\textcolor{b l a c k}{\sin}}} \beta \cos \left(- \beta\right) \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\frac{1}{\sin} \beta}}}$

$= \cos \left(- \beta\right)$

$= \cos \left(0 - \beta\right)$

$= \cos 0 \cos \beta + \sin 0 \sin \beta$

$= 1 \cdot \cos \beta + 0 \cdot \sin \beta$

$= 1 \cdot \cos \beta$

$= \cos \beta$

That's it. Hope this helped!