Problem 38 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{\cot y-\tan y}{\sin y \cos y}=\csc ^{2} y-\sec ^{2} y$$

Step-by-Step Solution

Answer

The identity holds true after simplification and substitution of identities.

Step 1: Express Cotangent and Tangent in Terms of Sine and Cosine

Recall the definitions of cotangent and tangent: \( \cot y = \frac{\cos y}{\sin y} \) and \( \tan y = \frac{\sin y}{\cos y} \). Substitute these into the left-hand side of the equation: \[\frac{\cot y - \tan y}{\sin y \cos y} = \frac{\frac{\cos y}{\sin y} - \frac{\sin y}{\cos y}}{\sin y \cos y}.\]

Step 2: Find a Common Denominator

Simplify the expression in the numerator by finding a common denominator for the fractions: \[\frac{\cos^2 y - \sin^2 y}{\sin y \cos y}.\]

Step 3: Recognize Trigonometric Identity for Difference of Squares

Notice that \( \cos^2 y - \sin^2 y \) can be rewritten using the identity \( \cos^2 y - \sin^2 y = (\cos y - \sin y)(\cos y + \sin y) \). Substitute this into the expression:\[\frac{(\cos y - \sin y)(\cos y + \sin y)}{\sin y \cos y}.\]

Step 4: Simplify the Expression

Simplify by dividing each term in the numerator by the denominator, resulting in:\[\frac{\cos y - \sin y}{\sin y} + \frac{\cos y + \sin y}{\cos y}.\] Convert these back to the trigonometric forms:\( \csc y - \sec y + \csc y + \sec y \).

Step 5: Use the Identity for Cosecant and Secant Squares

Recall the Pythagorean identities: \( \csc^2 y = 1 + \cot^2 y \) and \( \sec^2 y = 1 + \tan^2 y \). Thus:\[\csc^2 y - \sec^2 y = \cot^2 y - \tan^2 y.\]

Step 6: Equate and Verify Both Sides

Observe that:\[\cot^2 y - \tan^2 y = \frac{1}{\sin^2 y} - \frac{1}{\cos^2 y}\]which matches with the simplified left-hand side after conversions. Thus, the original equation holds true.

Key Concepts

Cotangent

The cotangent of an angle is a fundamental trigonometric function related to both sine and cosine. It's denoted as \( \cot y \) and is defined as the reciprocal of the tangent, giving us:

• \( \cot y = \frac{1}{\tan y} \)
• Alternatively, in terms of sine and cosine: \( \cot y = \frac{\cos y}{\sin y} \)

Cotangent is primarily used in trigonometric identities and equations where it helps to simplify expressions.
In the given exercise, by expressing cotangent in terms of sine and cosine, we easily manipulate and simplify trigonometric expressions, especially when combined with tangent.

Tangent

The tangent function, represented as \( \tan y \), is one of the primary trigonometric functions. It's vital for various applications in both mathematics and physics.

• The basic definition is \( \tan y = \frac{\sin y}{\cos y} \)
• It's the reciprocal of cotangent: \( \tan y = \frac{1}{\cot y} \)

Tangent is particularly useful in solving problems involving slopes and angles of elevation or depression.
In our exercise, transforming \( \tan y \) to its sine and cosine form allowed us to find a common denominator, simplifying the identity verification process.

Pythagorean Identities

Pythagorean identities form the backbone of many trigonometric simplifications and calculations. They relate the squares of the three primary trigs: sine, cosine, and tangent.

• The main identities are: \( \sin^2 y + \cos^2 y = 1 \)
• \( 1 + \tan^2 y = \sec^2 y \)
• \( 1 + \cot^2 y = \csc^2 y \)

These identities are crucial for transforming trigonometric terms into simpler forms or alternative expressions.
In the problem at hand, we used the identity \( \csc^2 y - \sec^2 y = \cot^2 y - \tan^2 y \) to equate both sides, showcasing their utility in verifying identities.

Cosecant

The cosecant function, abbreviated as \( \csc y \), is the reciprocal of the sine function. It's particularly useful for calculations involving right triangles and trigonometric identities.

• Defined as: \( \csc y = \frac{1}{\sin y} \)
• It plays a role in both the identity \( \csc^2 y = 1 + \cot^2 y \) and solving complex trigonometric expressions

Cosecant often appears in higher-level trigonometric problem-solving.
In our exercise, understanding \( \csc^2 y \) allowed us to leverage its identity for transforming and simplifying both sides of the equation effectively.

Secant

The secant function, denoted by \( \sec y \), is the reciprocal of the cosine function.

• Mathematically, it's expressed as \( \sec y = \frac{1}{\cos y} \)
• It's integral to the identity \( \sec^2 y = 1 + \tan^2 y \)

Secant is essential for complex trigonometric calculations, particularly in calculus and analytical geometry.
In our given example, knowing \( \sec^2 y \) and its identity was key in verifying the original equation, as it allowed transformation and comparison of both sides to prove they are equal.