Answer :
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]
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]
