Problem 16 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\cot x}{\csc x}=\cos x$$

Step-by-Step Solution

Answer

Question: Prove the identity \(\frac{\cot x}{\csc x}=\cos x\). Answer: To prove the identity, we first convert cotangent and cosecant to their basic trigonometric terms, simplify the equation, and cancel out any common terms. After completing these steps, we arrive at the equation \(\cos x = \cos x\), confirming the given identity is true.

Step 1: Convert cotangent and cosecant to basic trigonometric terms

To make it easier to work with the given equation, begin by converting the cotangent and cosecant terms into basic trigonometric terms, sine and cosine. Cotangent can be written as \(\cot x = \frac{\cos x}{\sin x}\) and cosecant can be written as \(\csc x = \frac{1}{\sin x}\). Substitute these terms into the original equation: $$\frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}} = \cos x$$

Step 2: Simplify the equation

Next, we can simplify the equation. To do so, we will multiply the numerator by the reciprocal of the denominator: $$\frac{\cos x}{\sin x} \cdot \frac{\sin x}{1} = \cos x$$

Step 3: Cancel out common terms

Now, we can cancel the common terms, sine, on both sides of the equation: $$\frac{\cos x}{\cancel{\sin x}} \cdot \cancel{\sin x} = \cos x$$ This leaves us with: $$\cos x = \cos x$$ Thus, we have successfully proven the identity: $$\frac{\cot x}{\csc x}=\cos x$$

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in understanding relationships in triangles and circles. They describe the ratios of sides in right-angled triangles and can also represent circular motion. These functions are sine, cosine, tangent, cotangent, secant, and cosecant. Each has its specific role and is derived from the geometric properties of triangles or unit circle properties.

• Sine (\( \sin \)): The ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos \)): The ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): The ratio of the opposite side to the adjacent side.
• Cotangent (\( \cot \)): The reciprocal of tangent.
• Secant (\( \sec \)): The reciprocal of cosine.
• Cosecant (\( \csc \)): The reciprocal of sine.

Together, these functions help us understand angles, periodicity, and wave patterns. By mastering their identities and interrelations, complex problems can be simplified and solved with ease.

Cotangent

Cotangent is one of the six main trigonometric functions often abbreviated as \( \cot \). It is defined as the reciprocal of the tangent function, or more simply: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \]This relation shows that cotangent is the ratio of the adjacent side over the opposite side in a right-angled triangle. You can also interpret cotangent through the unit circle, showing the relationship between cosine and sine.

• Importance: Cotangent is used to compute angles or solve equations where direct assessment of the sine or cosine isn't possible.
• Relevance: Cotangent appears in many mathematical applications, such as calculus and complex number theory.

Understanding cotangent's relationship with sine and cosine is crucial to mastering trigonometric identities and equations.

Cosecant

Cosecant is another significant trigonometric function, represented by \( \csc \). It is the reciprocal of the sine function. In formula terms, it's expressed as:\[ \csc x = \frac{1}{\sin x} \]This formula tells us that cosecant results from dividing one by the sine of an angle, which is typically used in various contexts where the sine value is non-zero. It uniquely ties into trigonometric identities and simplifies numerous trigonometric expressions.

• Usage: Cosecant proves useful in trigonometric identities, allowing transformations and simplifications in equations.
• Practical Applications: It's found in disciplines such as acoustics, signal processing, and other applied science fields.

Sine and Cosine Functions

Among the trigonometric functions, sine and cosine are the most commonly used. They are based on the fundamental properties of right-angled triangles and the unit circle.
The sine function, denoted \( \sin \), is defined as the ratio of the opposite side to the hypotenuse:\[ \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \]Similarly, the cosine function, or \( \cos \), represents the ratio of the adjacent side to the hypotenuse:\[ \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} \]These definitions hold true in the context of a right-angled triangle.

• Relation to Unit Circle: Both sine and cosine help describe the coordinates of a point on the unit circle.
• Periodicity: Sine and cosine have a periodic cycle, making them essential in modeling periodic phenomena.

Understanding these two key functions is critical as they form the foundation for more advanced concepts and are involved in many areas like physics, engineering, and signal processing.