To determine which expression is equivalent to [tex]\((1+\cos(x))^2 \tan\left(\frac{x}{2}\right)\)[/tex], let's go through the problem methodically.
1. Rewrite the original expression:
The original expression is:
[tex]\[
(1+\cos(x))^2 \tan\left(\frac{x}{2}\right)
\][/tex]
2. Identify potential equivalent expressions from the options:
a. [tex]\(\sin(x)\)[/tex]
b. [tex]\(1 - \cos(x)\)[/tex]
c. [tex]\(1 - \cos^2(x)\)[/tex]
d. [tex]\((1 + \cos(x)) \sin(x)\)[/tex]
3. Evaluate each of the potential equivalent expressions:
Since this is a trigonometric expression, let's consider simplifications and trigonometric identities that might help.
4. Methods for simplification:
- We can rewrite [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex] using half-angle identities:
[tex]\[
\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)}
\][/tex]
Substituting this into the given expression:
[tex]\[
(1 + \cos(x))^2 \cdot \frac{\sin(x)}{1 + \cos(x)}
\][/tex]
Simplify this expression:
[tex]\[
(1 + \cos(x)) \cdot \sin(x)
\][/tex]
5. Compare this result to the given options:
The simplified form we obtained is:
[tex]\[
(1 + \cos(x)) \sin(x)
\][/tex]
Now, compare this with the options provided:
a. [tex]\(\sin(x)\)[/tex]
b. [tex]\(1 - \cos(x)\)[/tex]
c. [tex]\(1 - \cos^2(x)\)[/tex]
d. [tex]\((1 + \cos(x)) \sin(x)\)[/tex]
We observe that our simplified expression [tex]\((1 + \cos(x)) \sin(x)\)[/tex] matches option d.
Therefore, the expression that is equivalent to [tex]\((1 + \cos(x))^2 \tan\left(\frac{x}{2}\right)\)[/tex] is:
[tex]\[
(1 + \cos(x)) \sin(x)
\][/tex]
So the correct answer is:
[tex]\[
(1 + \cos(x)) \sin(x)
\][/tex]
