# How do you simplify (sin theta csc theta)/cot theta?

Mar 11, 2018

color(green)(tan theta

#### Explanation:

From the above trigonometric identities,

$\csc \theta = \left(\frac{1}{\sin} \theta\right) , \cot \theta = \frac{1}{\tan} \theta = \cos \frac{\theta}{\sin} \theta$

Substituting the above in the given sum,

$\frac{\cancel{\sin} \theta \cdot \left(\frac{1}{\cancel{\sin}} \theta\right)}{\cos \frac{\theta}{\sin} \theta}$

=> sin theta / cos theta = color(green)(tan theta

Mar 11, 2018

$\frac{\sin \theta \csc \theta}{\cot} \theta = \tan \theta$.

#### Explanation:

Firstly, the cosecant function is defined as the inverse sine function:
$\csc \varphi = \frac{1}{\sin} \varphi$
and the cotangent function as the inverse tangent function: $\cot \varphi = \frac{1}{\tan} \varphi = \frac{1}{\sin \frac{\varphi}{\cos} \varphi} = \cos \frac{\varphi}{\sin} \varphi$.

Substituting it into the equation, we have

$\frac{\sin \theta \cdot \frac{1}{\textcolor{red}{\sin \theta}}}{\textcolor{red}{\cos \frac{\theta}{\sin} \theta}}$

We know that the product of a number and its inverse is always $1$ :

$\frac{\textcolor{red}{1}}{\textcolor{red}{\cos \frac{\theta}{\sin} \theta}} = \sin \frac{\theta}{\cos} \theta = \tan \theta$.