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.