Let's simplify the given expression: [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex].
1. First, recall that [tex]\(\sec x\)[/tex] is the reciprocal of [tex]\(\cos x\)[/tex], so [tex]\(\sec x = \frac{1}{\cos x}\)[/tex].
2. Substitute this into the expression:
[tex]\[
\frac{\sin x}{\frac{1}{\cos x} + 1}
\][/tex]
3. To combine the terms in the denominator, find a common denominator:
[tex]\[
\frac{\sin x}{\frac{1 + \cos x}{\cos x}}
\][/tex]
4. When dividing by a fraction, multiply by its reciprocal:
[tex]\[
\sin x \cdot \frac{\cos x}{1 + \cos x}
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{\sin x \cos x}{1 + \cos x}
\][/tex]
Given this step-by-step simplification process, we can conclude that the simplified form of the expression [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex] is indeed [tex]\(\frac{\sin x \cos x}{1 + \cos x}\)[/tex]. However, there is the possibility that we can simplify this further, but according to our computed values, that's the required simplified form, which matches with [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex].
