its tan over sec as a fraction for the integral

Ah, I see! You're referring to the integral where the integrand is the fraction \( \frac{\tan(x)}{\sec(x)} \). Let’s solve the following integral:

\[
\int \frac{\tan(x)}{\sec(x)} \, dx
\]

### Step 1: Simplify the expression

We know the relationship between tangent and secant:

\[
\sec(x) = \frac{1}{\cos(x)}, \quad \tan(x) = \frac{\sin(x)}{\cos(x)}
\]

So, the integrand simplifies as:

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

Thus, the integral becomes:

\[
\int \sin(x) \, dx
\]

### Step 2: Integrate

The integral of \( \sin(x) \) is a standard result:

\[
\int \sin(x) \, dx = -\cos(x) + C
\]

### Final Answer:
The solution to the integral is:

\[
\int \frac{\tan(x)}{\sec(x)} \, dx = -

Learn how to solve the integral of the fraction \( \frac{\tan(x)}{\sec(x)} \) with this detailed step-by-step solution. This problem covers core trigonometric concepts, simplifying the expression using identities, and finding the integral. Suitable for students in Grades 11-12.

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