Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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]

Sagot :

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]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.