Which expression is equivalent to [tex]\((1+\cos(x))^2 \left( \tan \left(\frac{x}{2}\right) \right) \)[/tex]?

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]



Answer :

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]