Simplify the following trigonometric expression:

[tex]\[ \frac{1}{\csc x + \cot x} + \frac{1}{\csc x - \cot x} \][/tex]

A. [tex]\(\csc x\)[/tex]

B. [tex]\(2 \csc x\)[/tex]



Answer :

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]