Multiply the numerator and denominator of the fraction by the conjugate of the denominator, and then simplify. (Write your answer to part (b) in terms of sine and/or cosine.)
1/1+cos x

To simplify the expression \(\frac{1}{1 + \cos x}\) by multiplying both the numerator and the denominator by the conjugate of the denominator, we proceed as follows:

### Step 1: Identify the conjugate of the denominator
The denominator is \(1 + \cos x\), and its conjugate is \(1 - \cos x\). 

### Step 2: Multiply the numerator and denominator by the conjugate
We multiply both the numerator and the denominator by \(1 - \cos x\):

\[
\frac{1}{1 + \cos x} \times \frac{1 - \cos x}{1 - \cos x} = \frac{1 - \cos x}{(1 + \cos x)(1 - \cos x)}
\]

### Step 3: Simplify the denominator
Use the difference of squares formula for the denominator:

\[
(1 + \cos x)(1 - \cos x) = 1^2 - (\cos x)^2 = 1 - \cos^2 x
\]

Thus, the expression becomes:

\[
\frac{1 - \cos x}{1 - \cos^2 x}
\]

### Step 4: Use the Pythagorean identity
We know that \(1 - \cos^2 x = \sin^2 x\) (Pythagorean identity). So the expression becomes:

\[
\frac{1 - \cos x}{\sin^2 x}
\]

### Final Answer:
\[
\frac{1 - \cos x}{\sin^2 x}
\]

This is the simplified form of \(\frac{1}{1 + \cos x}\) in terms of sine and cosine.

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Do you need further details or have any questions?

Here are some related questions:
1. What is the significance of using conjugates in simplifying expressions?
2. How does the difference of squares formula apply in this case?
3. Can the Pythagorean identity be derived from a geometrical perspective?
4. What happens when \(\cos x = 1\) or \(\cos x = -1\)?
5. Can this expression be further simplified for specific values of \(x\)?

**Tip**: When dealing with trigonometric identities, always keep Pythagorean identities in mind—they often simplify complex expressions!

Multiply the numerator and denominator of the fraction by the conjugate of the denominator, and then simplify. (Write your answer to part (b) in terms of sine and/or cosine.) 1/(1 + cos x)

Learn how to simplify 1/(1 + cos x) by multiplying by the conjugate of the denominator and applying trigonometric identities. This step-by-step solution covers the use of the difference of squares formula and the Pythagorean identity. Ideal for students in Grades 10-12.

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