How do I simplify sec x by tan x?

To simplify the expression \(\sec x\) divided by \(\tan x\), we can start by recalling the definitions of these trigonometric functions in terms of sine and cosine. This will help us rewrite the expression in a more manageable form.

Understanding the Functions

First, let's define \(\sec x\) and \(\tan x\):

• \(\sec x\) is the reciprocal of cosine, so \(\sec x = \frac{1}{\cos x}\).
• \(\tan x\) is the ratio of sine to cosine, thus \(\tan x = \frac{\sin x}{\cos x}\).

Rewriting the Expression

Now, we can substitute these definitions into the expression \(\frac{\sec x}{\tan x}\):

\[ \frac{\sec x}{\tan x} = \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} \]

Simplifying the Fraction

When we divide by a fraction, it’s the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as follows:

\[ \frac{1}{\cos x} \times \frac{\cos x}{\sin x} = \frac{1 \cdot \cos x}{\cos x \cdot \sin x} = \frac{1}{\sin x} \]

Final Result

Since \(\frac{1}{\sin x}\) is the definition of \(\csc x\), we can conclude that:

\[ \frac{\sec x}{\tan x} = \csc x \]

Summary

In summary, by using the definitions of secant and tangent, we simplified \(\frac{\sec x}{\tan x}\) to \(\csc x\). This process illustrates how understanding the relationships between trigonometric functions can make simplification straightforward. If you have any further questions or need clarification on any steps, feel free to ask!