Let's analyze the proof step by step to determine the expression that will complete step 3.
Given:
Step 1:
[tex]\[
\cos^2\left(\frac{x}{2}\right) = \frac{\sin(x) + \tan(x)}{2 \tan(x)}
\][/tex]
Step 2:
[tex]\[
\cos^2\left(\frac{x}{2}\right) = \frac{\sin(x) + \frac{\sin(x)}{\cos(x)}}{2 \left(\frac{\sin(x)}{\cos(x)}\right)}
\][/tex]
Next, we need to simplify the expression on the right-hand side to proceed to Step 3.
First, combine like terms in the numerator:
[tex]\[
\sin(x) + \frac{\sin(x)}{\cos(x)} = \frac{\sin(x) \cos(x) + \sin(x)}{\cos(x)}
\][/tex]
So, the numerator becomes:
[tex]\[
= \frac{\sin(x) \cos(x) + \sin(x)}{\cos(x)}
\][/tex]
Now, rewrite the entire fraction:
[tex]\[
\cos^2\left(\frac{x}{2}\right) = \frac{\frac{\sin(x) \cos(x) + \sin(x)}{\cos(x)}}{2 \left(\frac{\sin(x)}{\cos(x)}\right)}
\][/tex]
To simplify the right-hand side further, multiply numerator and denominator by [tex]\(\cos(x)\)[/tex]:
[tex]\[
= \frac{(\sin(x) \cos(x) + \sin(x)) \cdot \cos(x)}{2 \sin(x) \cdot \cos(x)}
\][/tex]
After canceling out the [tex]\(\cos(x)\)[/tex] terms, we get:
[tex]\[
= \frac{\sin(x) (\cos(x) + 1)}{2 \sin(x)}
\][/tex]
Lastly, cancel out [tex]\(\sin(x)\)[/tex] in the numerator and the denominator:
[tex]\[
= \frac{\cos(x) + 1}{2}
\][/tex]
We can see that this is the desired simplification.
Given the available options:
- [tex]\(\sin^2(x)\)[/tex]
- [tex]\(2 \sin(x)\)[/tex]
- [tex]\(2 \sin(x) \cos(x)\)[/tex]
- [tex]\(\sin(x) \cos(x) + \sin(x)\)[/tex]
The correct expression for step 3 in the proof to bridge from Step 2 to Step 4 is:
[tex]\[
\sin(x) \cos(x) + \sin(x)
\][/tex]
