To simplify the expression [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex], we need to recall some trigonometric identities and perform the simplification step-by-step. Let's start by analyzing the given expression.
1. Start with the given expression:
[tex]\[
\frac{\sin x}{\sec x + 1}
\][/tex]
2. Recall the definition of [tex]\(\sec x\)[/tex]:
[tex]\(\sec x = \frac{1}{\cos x}\)[/tex]
3. Substitute this identity into the expression:
[tex]\[
\frac{\sin x}{\frac{1}{\cos x} + 1}
\][/tex]
4. Find a common denominator in the denominator of our fraction:
[tex]\[
\frac{\sin x}{\frac{1 + \cos x}{\cos x}}
\][/tex]
5. Simplify the complex fraction by multiplying the numerator and the denominator by [tex]\(\cos x\)[/tex]:
[tex]\[
\frac{\sin x \cdot \cos x}{1 + \cos x}
\][/tex]
6. Recognize that this is the simplified form.
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{\sin x \cos x}{1 + \cos x}}
\][/tex]
