Review the proof.

\begin{tabular}{|c|c|}
\hline
Step & Statement \\
\hline
1 & [tex]$\cos ^2\left(\frac{x}{2}\right)=\frac{\sin (x)+\tan (x)}{2 \tan (x)}$[/tex] \\
\hline
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] \\
\hline
3 & [tex]$\cos ^2\left(\frac{x}{2}\right)=\frac{\frac{?}{\cos (x)}}{\frac{2 \sin (x)}{\cos (x)}}$[/tex] \\
\hline
4 & [tex]$\cos ^2\left(\frac{x}{2}\right)=\frac{\frac{(\sin (x))(\cos (x)+1)}{\cos (x)}}{\frac{2 \sin (x)}{\cos (x)}}$[/tex] \\
\hline
5 & [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] \\
\hline
6 & [tex]$\cos ^2\left(\frac{x}{2}\right)=\frac{\cos (x)+1}{2}$[/tex] \\
\hline
7 & [tex]$\cos \left(\frac{x}{2}\right)= \pm \sqrt{\frac{\cos (x)+1}{2}}$[/tex] \\
\hline
8 & [tex]$\cos \left(\frac{x}{2}\right)= \pm \sqrt{\frac{1+\cos (x)}{2}}$[/tex] \\
\hline
\end{tabular}

Which expression will complete step 3 in the proof?

A. [tex]$\sin ^2(x)$[/tex]

B. [tex]$2 \sin (x)$[/tex]

C. [tex]$2 \sin (x) \cos (x)$[/tex]

D. [tex]$\sin (x) \cos (x)+\sin (x)$[/tex]



Answer :

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]