Answer :
Let's analyze the correct derivation of [tex]\(\sin(2x)\)[/tex] using trigonometric identities.
### Step-by-Step Solution:
1. Expression for [tex]\(\sin(2x)\)[/tex]:
[tex]\[ \sin(2x) = \sin(x + x) \][/tex]
This is a correct starting point using the angle addition formula for sine.
2. Applying the Sine Angle Addition Formula:
According to the sine addition formula, [tex]\(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\)[/tex]. Applying this to [tex]\(\sin(x + x)\)[/tex],
[tex]\[ \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
This simplifies to:
[tex]\[ \sin(2x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
Since [tex]\(\sin(x)\cos(x)\)[/tex] and [tex]\(\cos(x)\sin(x)\)[/tex] are the same, we have:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
Therefore, Step 2 is correct as it stands.
3. Simplification of the Expression:
In the next step, we should derive the final simplified form.
[tex]\[ \sin(2x) \neq (\sin(x)\cos(x))^2 \][/tex]
The correct simplification we derived is:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
### Identifying the Error:
1. The expression in Step 2 is correct as it uses [tex]\(\sin(x)\cos(x) + \cos(x)\sin(x)\)[/tex].
2. The primary error lies in Step 3. Instead of squaring the product [tex]\(\sin(x)\cos(x)\)[/tex], the correct derivative from Step 2 should be [tex]\(\sin(2x) = 2\sin(x)\cos(x)\)[/tex].
Therefore:
- Step 2 is correctly:
[tex]\[ \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
- Step 3 should be corrected as:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
### Conclusion:
The correct choice indicating where the error is, corresponds to:
Step 3 should read [tex]\( = 2 \sin(x) \cos(x) \)[/tex].
### Step-by-Step Solution:
1. Expression for [tex]\(\sin(2x)\)[/tex]:
[tex]\[ \sin(2x) = \sin(x + x) \][/tex]
This is a correct starting point using the angle addition formula for sine.
2. Applying the Sine Angle Addition Formula:
According to the sine addition formula, [tex]\(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\)[/tex]. Applying this to [tex]\(\sin(x + x)\)[/tex],
[tex]\[ \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
This simplifies to:
[tex]\[ \sin(2x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
Since [tex]\(\sin(x)\cos(x)\)[/tex] and [tex]\(\cos(x)\sin(x)\)[/tex] are the same, we have:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
Therefore, Step 2 is correct as it stands.
3. Simplification of the Expression:
In the next step, we should derive the final simplified form.
[tex]\[ \sin(2x) \neq (\sin(x)\cos(x))^2 \][/tex]
The correct simplification we derived is:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
### Identifying the Error:
1. The expression in Step 2 is correct as it uses [tex]\(\sin(x)\cos(x) + \cos(x)\sin(x)\)[/tex].
2. The primary error lies in Step 3. Instead of squaring the product [tex]\(\sin(x)\cos(x)\)[/tex], the correct derivative from Step 2 should be [tex]\(\sin(2x) = 2\sin(x)\cos(x)\)[/tex].
Therefore:
- Step 2 is correctly:
[tex]\[ \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x) \][/tex]
- Step 3 should be corrected as:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
### Conclusion:
The correct choice indicating where the error is, corresponds to:
Step 3 should read [tex]\( = 2 \sin(x) \cos(x) \)[/tex].
