To simplify the trigonometric expression
[tex]\[ \frac{1}{\csc x + \cot x} + \frac{1}{\csc x - \cot x}, \][/tex]
we need to manipulate and simplify each term step-by-step.
First, let's recall the definitions:
- [tex]\(\csc x = \frac{1}{\sin x}\)[/tex]
- [tex]\(\cot x = \frac{\cos x}{\sin x}\)[/tex]
Let's simplify the expression term-by-term:
### Step 1: Simplify Each Fraction
1. For the first fraction:
[tex]\[ \frac{1}{\csc x + \cot x} \][/tex]
Using the definitions of [tex]\(\csc x\)[/tex] and [tex]\(\cot x\)[/tex]:
[tex]\[ \frac{1}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}} \][/tex]
Combine the terms in the denominator over a common denominator:
[tex]\[ \frac{1}{\frac{1 + \cos x}{\sin x}} \][/tex]
Invert the fraction:
[tex]\[ \frac{\sin x}{1 + \cos x} \][/tex]
2. For the second fraction:
[tex]\[ \frac{1}{\csc x - \cot x} \][/tex]
Using the definitions of [tex]\(\csc x\)[/tex] and [tex]\(\cot x\)[/tex]:
[tex]\[ \frac{1}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}} \][/tex]
Combine the terms in the denominator over a common denominator:
[tex]\[ \frac{1}{\frac{1 - \cos x}{\sin x}} \][/tex]
Invert the fraction:
[tex]\[ \frac{\sin x}{1 - \cos x} \][/tex]
### Step 2: Add the Two Simplified Fractions
Now, add the two fractions we have obtained:
[tex]\[ \frac{\sin x}{1 + \cos x} + \frac{\sin x}{1 - \cos x} \][/tex]
To add these fractions, we combine them over a common denominator. The common denominator is [tex]\((1 + \cos x)(1 - \cos x)\)[/tex].
[tex]\[ \frac{\sin x (1 - \cos x) + \sin x (1 + \cos x)}{(1 + \cos x)(1 - \cos x)} \][/tex]
Simplify the numerator:
[tex]\[ \sin x (1 - \cos x) + \sin x (1 + \cos x) \][/tex]
Factor out [tex]\(\sin x\)[/tex]:
[tex]\[ \sin x [ (1 - \cos x) + (1 + \cos x) ] \][/tex]
Combine the terms within the brackets:
[tex]\[ \sin x [ 1 - \cos x + 1 + \cos x ] \][/tex]
[tex]\[ \sin x [ 2 ] \][/tex]
[tex]\[ 2 \sin x \][/tex]
So, the numerator simplified is [tex]\(2 \sin x\)[/tex].
For the denominator, using the difference of squares:
[tex]\[ (1 + \cos x)(1 - \cos x) = 1 - \cos^2 x \][/tex]
We know from the Pythagorean identity that:
[tex]\[ 1 - \cos^2 x = \sin^2 x \][/tex]
So the denominator simplifies to [tex]\(\sin^2 x\)[/tex].
### Step 3: Combine the Simplified Numerator and Denominator
Putting it all together, the simplified expression is:
[tex]\[ \frac{2 \sin x}{\sin^2 x} \][/tex]
Simplify by cancelling out [tex]\(\sin x\)[/tex] in the numerator and denominator:
[tex]\[ \frac{2}{\sin x} \][/tex]
Recall that [tex]\(\frac{1}{\sin x} = \csc x\)[/tex]:
[tex]\[ \frac{2}{\sin x} = 2 \csc x \][/tex]
Therefore, the simplified expression is:
[tex]\[ 2 \csc x \][/tex]
