Use the fundamental identities and appropriate algebraic operations to simplify the expression.

StartFraction 1 Over 1 minus sine x EndFraction minus StartFraction 1 Over 1 plus sine x EndFraction11−sin x−11+sin x

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Part 1

Choose the correct answer below.

To simplify the expression 

\[
\frac{1}{1 - \sin x} - \frac{1}{1 + \sin x}
\]

we can follow these steps:

### Step 1: Find a Common Denominator
The common denominator for the two fractions is \( (1 - \sin x)(1 + \sin x) \).

### Step 2: Rewrite Each Fraction with the Common Denominator
Using the common denominator, we have:

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

### Step 3: Substitute and Simplify the Expression
Now, we can rewrite the expression as:

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

Combine the fractions:

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

### Step 4: Simplify the Numerator
Expand the terms in the numerator:

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

Combine like terms in the numerator:

\[
= \frac{2 \sin x}{(1 - \sin x)(1 + \sin x)}
\]

### Step 5: Apply the Difference of Squares in the Denominator
Using the identity \( (1 - \sin x)(1 + \sin x) = 1 - \sin^2 x \), rewrite the denominator:

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

### Step 6: Use the Pythagorean Identity
Recall that \( 1 - \sin^2 x = \cos^2 x \). Substitute this in:

\[
= \frac{2 \sin x}{\cos^2 x}
\]

### Step 7: Separate the Terms
Rewrite the expression as:

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

### Step 8: Simplify Using Trigonometric Identities
Now, use \( \frac{\sin x}{\cos x} = \tan x \) and \( \frac{1}{\cos x} = \sec x \):

\[
= 2 \tan x \sec x
\]

### Final Answer:
\[
\frac{1}{1 - \sin x} - \frac{1}{1 + \sin x} = 2 \tan x \sec x
\]

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### Would you like more details on any steps?

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### 5 Related Questions:
1. How would you simplify \( \frac{1}{1 - \cos x} - \frac{1}{1 + \cos x} \)?
2. What is the trigonometric identity for \( \sin^2 x + \cos^2 x \)?
3. How can you express \( \tan x \sec x \) in terms of \( \sin x \) and \( \cos x \)?
4. How would the expression change if we used \( \cot x \) instead of \( \tan x \)?
5. Can this technique be applied to simplify \( \frac{1}{1 - \tan x} - \frac{1}{1 + \tan x} \)?

### Tip:
When simplifying trigonometric expressions, using common identities (like Pythagorean and reciprocal identities) can make complex expressions much easier to work with.

Use the fundamental identities and appropriate algebraic operations to simplify the expression 1/(1 - sin x) - 1/(1 + sin x)

Master the simplification of the trigonometric expression 1/(1 - sin x) - 1/(1 + sin x) with step-by-step guidance. This example uses fundamental identities, including the Pythagorean identity and difference of squares, to simplify the expression to 2 tan x sec x. Ideal for students in Grades 10-12.

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