Problem 31 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{1}{\cos x + 1} + \dfrac{1}{\cos x - 1} = -2 \csc x \cot x \)

Step-by-Step Solution

Answer

Therefore, \( \dfrac{1}{\cos x + 1} + \dfrac{1}{\cos x - 1} = -2 \csc x \cot x \)

Step 1: Rewrite the fractions

The first step is to add the two fractions on the left-hand side. The resultant fraction will be \(\dfrac{\cos x - 1 + \cos x + 1}{\cos^2 x - 1}\), which simplifies further to \(\dfrac{2 \cos x}{\cos^2 x - 1}\). This comes from having a common denominator.

Step 2: Apply Pythagorean identities

The second step is to apply the Pythagorean identities to the denominator, which results in \(\dfrac{2 \cos x}{- \sin^2 x}\). Remember that the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) translates into \(\cos^2 x - 1 = - \sin^2 x\).

Step 3: Simplify

Now, you can further simplify the above expression to \( -2 \frac{\cos x}{\sin^2 x}\) .

Step 4: Rewrite in terms of cotangent and cosecant

Finally, rewrite the above expression in terms of cotangent and cosecant. The final result will be \(-2 \csc x \cot x\).

Key Concepts

Cosecant

Cosecant is a trigonometric function denoted as \( \csc x \). It is the reciprocal of the sine function, meaning it is equal to the inverse of sine. So, if you have an angle \( x \), the cosecant is calculated as:

• \( \csc x = \frac{1}{\sin x} \)

Understanding cosecant is crucial when working with reciprocal identities and in transforming expressions that involve sine.
It appears often in trigonometric identities and equations, such as the exercise you've worked on. When you rewrite terms using cosecant, you're essentially simplifying the expression into a form that may be easier to work with. Additionally, when you see \( \csc x \) in an equation, you can immediately understand that it involves the sine of \( x \) as a core component.

Cotangent

Cotangent is another important trigonometric function. It is the reciprocal of the tangent function, and often used in various trigonometric identities. The formula for cotangent is:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

You can see from the definition that cotangent relates to both sine and cosine,
which is why it frequently appears in the simplification and verification of identities involving these functions.
Cotangent helps in expressing angles and solving triangles. When simplified expressions like \( -2 \csc x \cot x \) show up, recognizing that \( \cot x \) is essentially \( \frac{\cos x}{\sin x} \) is key to understanding how the terms relate to each other.

Pythagorean Identities

Pythagorean identities are foundational relationships in trigonometry that express the interrelation between sine, cosine, and tangent. One of the most commonly used Pythagorean identities is:

• \( \sin^2 x + \cos^2 x = 1 \)

From this, we can derive other forms such as:

• \( \cos^2 x - 1 = -\sin^2 x \)
• \( 1 - \sin^2 x = \cos^2 x \)

These identities are powerful tools for simplifying complex trigonometric expressions by substituting terms,
and allow us to transform one form into another easily. For example, in the problem, using \( \cos^2 x - 1 = -\sin^2 x \) simplifies the expression, revealing a direct relationship between cosine and sine. This helps in further solving or proving identities, as it did in your exercise where \( \dfrac{2 \cos x}{- \sin^2 x} \) led to \( -2 \csc x \cot x \), simplifying what initially appeared complicated.