Problem 16 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{1}{\csc y-\cot y}=\csc y+\cot y$$

Step-by-Step Solution

Answer

Identity is verified: both sides match.

Step 1: Rewrite the Left Side of the Equation

We start with the left side of the equation, \( \frac{1}{\csc y - \cot y} \). To simplify this expression, note that \( \csc y = \frac{1}{\sin y} \) and \( \cot y = \frac{\cos y}{\sin y} \). Thus, we can rewrite the expression as \( \frac{1}{\frac{1}{\sin y} - \frac{\cos y}{\sin y}} \) which simplifies to \( \frac{1}{\frac{1 - \cos y}{\sin y}} \). Since dividing by a fraction is equivalent to multiplying by its reciprocal, this becomes \( \frac{\sin y}{1 - \cos y} \).

Step 2: Simplify Using Trigonometric Identities

We need to simplify \( \frac{\sin y}{1 - \cos y} \). We can use the identity \( 1 - \cos y = (\sin y)^2/(1 + \cos y) \) to rewrite the expression. To make this simplification useful, let's multiply the numerator and the denominator by the conjugate of the denominator, \( 1 + \cos y \), which gives us: \( \frac{\sin y (1 + \cos y)}{(1 - \cos y)(1 + \cos y)} \). The denominator simplifies to \( 1 - (\cos y)^2 = (\sin y)^2 \).

Step 3: Further Simplify the Expression

After substituting and simplifying, our expression becomes \( \frac{\sin y (1 + \cos y)}{(\sin y)^2} \). Cancel \( \sin y \) in the numerator and denominator to get \( \frac{1 + \cos y}{\sin y} \).

Step 4: Recognize the Expression as the Right Side

Finally, recognize that \( \frac{1 + \cos y}{\sin y} \) can be split into \( \frac{1}{\sin y} + \frac{\cos y}{\sin y} \), which is precisely \( \csc y + \cot y \). Hence, we have shown that the original expression simplifies exactly to the right side of the equation, verifying that \( \frac{1}{\csc y - \cot y} = \csc y + \cot y \).

Key Concepts

Understanding Cosecant

Cosecant, often represented as \(\csc\), is one of the six fundamental trigonometric functions. It's particularly important because it is the reciprocal of the sine function.

• If \( \sin y = \frac{opposite}{hypotenuse} \), then \( \csc y = \frac{hypotenuse}{opposite} \).
• This essentially means that \( \csc y = \frac{1}{\sin y} \).

Understanding cosecant in terms of its reciprocal relationship with sine helps in various trigonometric simplifications and solving trigonometric equations. It's typically used in cases where the sine of an angle is quite small, which makes the cosecant function larger and more significant. In the context of our exercise, recognizing \( \csc y = \frac{1}{\sin y} \) was crucial to converting terms into forms that are easy to manipulate and simplify.

Exploring Cotangent

Cotangent, denoted as \( \cot \), is another trigonometric function, often used in tandem with cosecant and tangent.

• It's defined as the reciprocal of tangent: \( \cot y = \frac{1}{\tan y} \).
• In terms of sine and cosine, \( \cot y = \frac{\cos y}{\sin y} \).

Cotangent is particularly useful when dealing with complementary angles and can often appear in problems involving right triangles. For instance, in the problem we solved, knowing \( \cot y \) as a fraction of cosine and sine was critical because it allowed us to rewrite \( \cot y \) in terms of known values, aiding in simplifying the given identity.

Reciprocal Identities Role

Reciprocal identities are fundamental relationships in trigonometry that express the relationships between trigonometric functions and their reciprocals. The primary reason these identities are valuable is that they allow us to switch between a function and its reciprocal, facilitating easier algebraic manipulation.

• \( \csc y = \frac{1}{\sin y} \)
• \( \sec y = \frac{1}{\cos y} \)
• \( \cot y = \frac{1}{\tan y} \)

By applying reciprocal identities in the provided exercise, we translated complex expressions into manageable components. This kind of transformation is an essential first step in verifying trigonometric identities or solving trigonometric equations effectively.

Trigonometric Simplification Techniques

Trigonometric simplification is a process of reducing complex trigonometric expressions into more manageable or familiar forms. This often involves using trig identities to rewrite parts of the expression, such as the Pythagorean identities, angle sum and difference identities, and, notably, reciprocal identities. A common technique is to multiply the numerator and the denominator by conjugates, as seen in our problem when we multiplied by \( 1 + \cos y \).

• This steps to utilize \( (a-b)(a+b) = a^2 - b^2 \), which simplifies expressions significantly.
• Factoring and canceling common terms are also widespread methods to simplify.

Using these techniques, the exercise solved \( \frac{1}{\csc y - \cot y} \) to show it was equivalent to \( \csc y + \cot y \), demonstrating how layered trig expressions can be resolved into straightforward identities.