Problem 66 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\sin A}{1-\cos A}-\cot A=\csc A$$

Step-by-Step Solution

Answer

The identity is verified; left side equals \( \csc A \).

Step 1: Rewriting the Left Side

Let's first simplify the left side of the equation. The expression is \( \frac{\sin A}{1-\cos A} - \cot A \). We know \( \cot A = \frac{\cos A}{\sin A} \). So, substitute \( \cot A \) in the equation:\[\frac{\sin A}{1-\cos A} - \frac{\cos A}{\sin A}\]

Step 2: Finding a Common Denominator

Now, to combine the fractions, we need a common denominator. The denominators are \((1-\cos A)\) and \(\sin A\). The common denominator is \(\sin A(1-\cos A)\). Rewrite each fraction with the common denominator:\[\frac{\sin A \cdot \sin A}{\sin A(1-\cos A)} - \frac{\cos A (1-\cos A)}{\sin A(1-\cos A)}\]

Step 3: Combining the Fractions

Combine the fractions on the left side:\[\frac{\sin^2 A - \cos A (1-\cos A)}{\sin A (1-\cos A)}\]Simplify the numerator:\[\sin^2 A - \cos A + \cos^2 A\]

Step 4: Using Pythagorean Identity

Apply the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\) to the numerator:\[1 - \cos A\]This simplifies the numerator to \(1 - \cos A\). Therefore, the fraction reduces to:\[\frac{1 - \cos A}{\sin A(1-\cos A)}\]

Step 5: Canceling Terms

Since \(1 - \cos A\) appears in the numerator and is part of the denominator, we can cancel it out:\[\frac{1}{\sin A}\]This results in \(\csc A\), which is the right side of the original identity.

Step 6: Verification Complete

We have shown that the left side \( \frac{\sin A}{1-\cos A} - \cot A \) simplifies to the right side \( \csc A \). Thus, the identity is verified.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc A \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means that:

• \( \csc A = \frac{1}{\sin A} \)

What's interesting about the cosecant is that it helps in understanding the relationship between the sides of a triangle and its angles in a different way compared to the sine function.
Remember, since \( \sin A = \frac{opposite}{hypotenuse} \) in a right triangle, \( \csc A \) becomes \( \frac{hypotenuse}{opposite} \). Thus, it can help determine the length of the hypotenuse when the angle and the length of the opposite side are known. However, the cosecant is not defined for angles where the sine of the angle is zero, in these instances it leads to undefined values.

Cotangent Function

The cotangent function, abbreviated as \( \cot A \), is another important trigonometric function. It is the reciprocal of the tangent function, defined as:

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

In right-angle trigonometry, cotangent represents the ratio of the adjacent side to the opposite side.
This means \( \cot A = \frac{adjacent}{opposite} \), which inversely relates to the concept of tangent. Understanding this reciprocal relationship is crucial in simplifying trigonometric identities and equations, just as we have done in the given exercise.
The ability to convert between \( \tan A \) and \( \cot A \) empowers you to manipulate expressions to verify identities or solve trigonometric equations more effectively.

Pythagorean Identity

One of the fundamental identities in trigonometry is the Pythagorean identity. This involves the sine and cosine functions and is expressed as:

• \( \sin^2 A + \cos^2 A = 1 \)

This identity is pivotal because it relates the squares of sine and cosine of any angle to one, showing they always equal parts of a unit circle.
In our exercise, this identity was used to simplify the numerator of the fraction. We transformed \( \sin^2 A - \cos A + \cos^2 A \) into \( 1 - \cos A \).
When simplifying trigonometric expressions, recognizing such identities allows you to convert complex expressions into simpler forms, making calculations and verifications much more manageable. It's one of the reasons why memorizing these identities can be very beneficial.

Common Denominator

Combining fractions is a common mathematical operation that often requires finding a common denominator. In trigonometry, this is particularly useful when dealing with expressions involving complex fractions like in our exercise:

• The original denominators were \( 1 - \cos A \) and \( \sin A \).
• Their common denominator is \( \sin A(1 - \cos A) \).

Having a common denominator allows us to combine separate terms into a single expression, facilitating further simplification.
This method proves particularly helpful in trigonometry, where combining and simplifying fractions can reveal underlying trigonometric identities, as demonstrated by turning the left side of the original identity into a single fraction. This step is often necessary to uncover the structure needed to utilize identities like the Pythagorean identity.