Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\cot \theta}{\csc \theta}=\cos \theta\)

Step-by-Step Solution

Answer

The equation is an identity, as both sides simplify to \( \cos \theta \).

Step 1: Express in terms of sine and cosine

Start by converting the trigonometric functions into terms of sine and cosine. We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute expressions

Substitute these expressions back into the left-hand side of the equation to get: \( \frac{\frac{\cos \theta}{\sin \theta}}{\frac{1}{\sin \theta}} \).

Step 3: Simplify the fraction

Simplify the fraction by multiplying the numerator by the reciprocal of the denominator: \( \frac{\cos \theta}{\sin \theta} \times \sin \theta \). The \( \sin \theta \) in the numerator and denominator cancel each other out.

Step 4: Result of simplification

After canceling the \( \sin \theta \), we are left with \( \cos \theta \), which is the right-hand side of the equation.

Step 5: Conclusion: Verify the equation

Since both sides of the equation are equal, the equation \( \frac{\cot \theta}{\csc \theta} = \cos \theta \) is verified as an identity.

Key Concepts

Understanding Cotangent

Cotangent, often abbreviated as "cot," is a fundamental trigonometric function. It is traditionally defined as the reciprocal of the tangent function.

• Mathematically, this is expressed as: \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
• Cotangent deals with the ratio of the adjacent side to the opposite side in a right triangle.
• As a reciprocal identity, cotangent is essential in simplifying complex trigonometric equations.

The clear understanding of cotangent's relationship to sine and cosine allows for smooth transitions when verifying or manipulating identities, such as the one in the provided exercise. Knowing this is useful for both solving trigonometric equations and understanding angular relationships in geometry.

Exploring Cosecant

Cosecant, abbreviated as "csc," is another key trigonometric function, known as the reciprocal of sine.

• Its definition can be stated as: \( \csc \theta = \frac{1}{\sin \theta} \).
• The cosecant function is crucial in equations like your exercise, often used to transform identities into simpler known forms.
• Similar to cotangent, it flips the relationship defined by sine, turning opposite over hypotenuse into hypotenuse over opposite in triangle terms.

By understanding the role of cosecant in equations, like converting it to sine equivalents, one can simplify and solve identities effectively. It's one of the less commonly used functions, but in some cases like this exercise, it allows for elegant solutions of trigonometric problems.

The Significance of Cosine

Cosine, or \( \cos \), is one of the most commonly used trigonometric functions.

• It is defined simply as the ratio of the adjacent side over the hypotenuse of a right triangle: \( \cos \theta = \frac{adjacent}{hypotenuse} \).
• As seen in the exercise, it often represents the simplified form of more complex trigonometric equations.
• One of the reasons for this is its property as part of the primary trigonometric identities, such as the Pythagorean identity: \( \cos^2 \theta + \sin^2 \theta = 1 \).

Cosine is integral in simplifying expressions and is frequently encountered when verifying identities. Once cotangent and cosecant were expressed in terms of sine and cosine, the relation \( \frac{\cot \theta}{\csc \theta} \) simplifies directly to \( \cos \theta \), demonstrating the practical importance of expressing complex equations through basic functions like cosine.