Great, let's review the proof step-by-step and determine the expression needed to complete step 3.
We start with the statement from step 1:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\frac{\sin(x) + \tan(x)}{2 \tan(x)}
\][/tex]
Next, in step 2, we replace [tex]\(\tan(x)\)[/tex] with [tex]\(\frac{\sin(x)}{\cos(x)}\)[/tex]:
[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]
To further simplify step 2, we need to combine the terms in the numerator. We can write:
[tex]\[
\sin(x) + \frac{\sin(x)}{\cos(x)} = \sin(x) \left(1 + \frac{1}{\cos(x)}\right)
\][/tex]
Now, express the combined numerator more compactly:
[tex]\[
\sin(x) \left(1 + \frac{1}{\cos(x)}\right) = \sin(x) \left(\frac{\cos(x) + 1}{\cos(x)}\right)
\][/tex]
This leads us to:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\frac{\sin(x) \left(\frac{\cos(x) + 1}{\cos(x)}\right)}{2\left(\frac{\sin(x)}{\cos(x)}\right)}
\][/tex]
Now we simplify the fraction, keeping the denominators unified:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\frac{\frac{\sin(x)(\cos(x) + 1)}{\cos(x)}}{\frac{2\sin(x)}{\cos(x)}}
\][/tex]
In step 4, this becomes:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\frac{\frac{\sin(x)(\cos(x) + 1)}{\cos(x)}}{\frac{2\sin(x)}{\cos(x)}}
\][/tex]
Next, we cancel out the [tex]\(\cos(x)\)[/tex] terms, and simplify:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\left(\frac{\sin(x)(\cos(x) + 1)}{\cos(x)}\right) \left(\frac{\cos(x)}{2 \sin(x)}\right)
\][/tex]
This results in:
[tex]\[
\cos^2\left(\frac{x}{2}\right)=\frac{\cos(x) + 1}{2}
\][/tex]
Finally, taking the square root:
[tex]\[
\cos\left(\frac{x}{2}\right)= \pm \sqrt{\frac{\cos(x) + 1}{2}}
\][/tex]
Matching this with our need to determine step 3, we see that it should be:
[tex]\[
\sin(x) \cos(x) + \sin(x)
\][/tex]
Thus, the expression that completes step 3 is:
[tex]\[
\sin(x) \cos(x) + \sin(x)
\][/tex]
