Question

How do I simplify $\frac{\sec x}{\tan x}$.

Answer 1

Simplify the given trigonometric expression :

Given expression: $\frac{\sec \left(x\right)}{\tan \left(x\right)}$.

It is known that $\sec \left(x\right)=\frac{1}{\cos \left(x\right)}$ and $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$.

Thus, $\frac{\sec \left(x\right)}{\tan \left(x\right)}=\frac{{\displaystyle \frac{1}{\cos \left(x\right)}}}{{\displaystyle \frac{\sin \left(x\right)}{\cos \left(x\right)}}}$

$$\begin{gathered} \Rightarrow \frac{\sec \left(x\right)}{\tan \left(x\right)}=\frac{1}{\cos \left(x\right)}\times \frac{\cos \left(x\right)}{\sin \left(x\right)} \\ \Rightarrow \frac{\sec \left(x\right)}{\tan \left(x\right)}=\frac{1}{\sin \left(x\right)} \end{gathered}$$

It is also known that, $\frac{1}{\sin \left(x\right)}=\mathrm{cosec}\left(x\right)$.

Hence, $\frac{\sec \left(x\right)}{\tan \left(x\right)}=\mathrm{cosec}\left(x\right)$.

Therefore, $\frac{\sec \left(\mathrm{x}\right)}{\tan \left(\mathrm{x}\right)}$ can be simplified as $\mathrm{cosec}\left(x\right)$.