To simplify the given expression [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex], let's proceed step-by-step:
1. Rewrite secant in terms of sine and cosine:
[tex]\[
\sec x = \frac{1}{\cos x}
\][/tex]
So the given expression becomes:
[tex]\[
\frac{\sin x}{\frac{1}{\cos x} + 1}
\][/tex]
2. Combine the terms in the denominator:
[tex]\[
\frac{1}{\cos x} + 1 = \frac{1 + \cos x}{\cos x}
\][/tex]
3. Rewrite the fraction:
[tex]\[
\frac{\sin x}{\frac{1 + \cos x}{\cos x}} = \sin x \cdot \frac{\cos x}{1 + \cos x}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{\sin x \cos x}{1 + \cos x}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex] is:
[tex]\[
\frac{\sin x \cos x}{1 + \cos x}
\][/tex]
The correct choice from the provided options is:
[tex]\[
\sin x \cos x + \sin x
\][/tex]
However, it seems there might be a discrepancy in interpretation of the options, as the simplified result:
[tex]\[
\frac{\sin x \cos x}{1 + \cos x}
\][/tex]
is the correct mathematical simplification. The provided options might need to be verified for correctness based on this accurate simplified result.
