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Review the proof.

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

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]

Sagot :

GinnyAnswer
To determine which expression completes step 3 in the proof, let’s start from step 2 and progress logically to match the required transformation.

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]

Let's simplify the right-hand side expression:

1. Combine the terms in the numerator:
[tex]\[ \frac{\sin(x) + \frac{\sin(x)}{\cos(x)}}{2 \left(\frac{\sin(x)}{\cos(x)}\right)} \][/tex]

Since [tex]\(\frac{\sin(x)}{\cos(x)} = \tan(x)\)[/tex], the numerator becomes:
[tex]\[ \sin(x) + \frac{\sin(x)}{\cos(x)} = \sin(x) + \sin(x) \cdot \frac{1}{\cos(x)} \][/tex]

Combine the fraction in the numerator:
[tex]\[ \frac{\sin(x) \cos(x) + \sin(x)}{\cos(x)} \][/tex]

2. Simplify the denominator:
[tex]\[ 2 \left(\frac{\sin(x)}{\cos(x)}\right) = \frac{2 \sin(x)}{\cos(x)} \][/tex]

3. Therefore, the entire fraction simplifies as follows:
[tex]\[ \cos^2\left(\frac{x}{2}\right) = \frac{\frac{\sin(x)\cos(x) + \sin(x)}{\cos(x)}}{\frac{2 \sin(x)}{\cos(x)}} \][/tex]

When you divide by a fraction, it's the same as multiplying by the reciprocal:
[tex]\[ \frac{\frac{\sin(x)\cos(x) + \sin(x)}{\cos(x)}}{\frac{2 \sin(x)}{\cos(x)}} = \left(\frac{\sin(x)\cos(x) + \sin(x)}{\cos(x)} \right) \cdot \left(\frac{\cos(x)}{2 \sin(x)}\right) \][/tex]

4. After simplification, we get:
[tex]\[ \cos^2\left(\frac{x}{2}\right) = \left(\frac{(\sin(x)\cos(x) + \sin(x)) \cdot \cos(x)}{\cos(x) \cdot 2 \sin(x)}\right) \][/tex]

5. Calculate the terms in the fraction:
[tex]\[ \cos^2\left(\frac{x}{2}\right) = \frac{(\sin(x)\cos(x) + \sin(x)) \cdot \cos(x)}{2 \sin(x) \cdot \cos(x)} \][/tex]

Since [tex]\(\cos(x)\)[/tex] cancels out:
[tex]\[ \cos^2\left(\frac{x}{2}\right) = \frac{\sin(x) \cos(x) + \sin(x)}{2 \sin(x)} \][/tex]

6. Finally:
[tex]\[ \cos^2\left(\frac{x}{2}\right) = \frac{\sin(x) (\cos(x) + 1)}{2 \sin(x)} = \frac{\cos(x)+1}{2} \][/tex]

At step 3, the expression inside the numerator is:
[tex]\[ \sin(x)\cos(x) + \sin(x) \][/tex]

Therefore, the expression that will complete step 3 in the proof is:
[tex]\[ \boxed{\sin(x) \cos(x) + \sin(x)} \][/tex]
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