Problem 10 - Contemporary Precalculus

Question

Prove the identity. $$\cot x \sin x=\cos x$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity: \(\cot x \sin x = \cos x\) Answer: To prove this identity, we first rewrite the cotangent function in terms of sine and cosine as follows: \(\cot x \sin x = \frac{\cos x}{\sin x} \sin x\). Next, we simplify the expression by cancelling out the sine terms in the numerator and denominator: \(\frac{\cos x}{\sin x} \sin x = \cos x \cdot \frac{\sin x}{\sin x}\). Since \(\frac{\sin x}{\sin x} = 1\), the expression becomes \(\cos x \cdot 1 = \cos x\), which matches the right side of the given identity. Therefore, we have proven that \(\cot x \sin x = \cos x\).

Step 1: Write the cotangent in terms of sine and cosine

Recall that the cotangent function, \(\cot x\), is defined as the quotient of the cosine by the sine function, \(\cot x = \frac{\cos x}{\sin x}\). Using this definition, we can rewrite the left side of the equation: $$ \cot x \sin x = \frac{\cos x}{\sin x} \sin x $$

Step 2: Simplify the expression

Now, we'll simplify the expression by cancelling out the sine terms in the numerator and denominator: $$ \frac{\cos x}{\sin x} \sin x = \cos x \cdot \frac{\sin x}{\sin x} $$ Since \(\frac{\sin x}{\sin x} = 1\), the expression becomes: $$ \cos x \cdot 1 = \cos x $$

Step 3: Compare the simplified expression with the right side of the equation

After simplifying the left side of the equation, we obtained \(\cos x\). This matches the right side of the given identity. Therefore, we have successfully proven the trigonometric identity: $$ \cot x \sin x = \cos x $$

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, particularly in the study of triangles and periodic phenomena. They relate the angles of a triangle to the ratios of its sides. The primary trigonometric functions are sine, cosine, and tangent, often abbreviated as \( \sin x \), \( \cos x \), and \( \tan x \) respectively. These functions are fundamental in trigonometry, which has applications in fields ranging from physics to engineering.Understanding these functions:

• Sine (\( \sin x \)) - This function represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos x \)) - This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan x \)) - It is the ratio of the opposite side to the adjacent side.

Trigonometric identities, like the one in the exercise, showcase the inter-convertibility of these functions. These identities form the backbone of simplifying expressions and solving trigonometric equations.

Cotangent Function

The cotangent function is one of the reciprocal trigonometric functions. It can be defined in terms of the sine and cosine functions. Specifically, the cotangent of an angle \( x \) is the reciprocal of the tangent function, or the ratio of the cosine to the sine: \[ \cot x = \frac{\cos x}{\sin x} \]This identity can be quite useful for simplifying expressions that involve trigonometric functions, particularly when you're trying to eliminate fractions or simplify complex trigonometric equations. Understanding how to transform trigonometric functions into one another by using these identities allows for greater flexibility in solving problems. It is crucial to memorize key identities like the one used here, as they are commonly used in calculus and analytic geometry.

Simplifying Expressions

Simplifying trigonometric expressions is an important skill that helps in making complex problems more manageable. In the given exercise, the aim was to prove an identity through simplification. By transforming \( \cot x \) into \( \frac{\cos x}{\sin x} \), we utilized the properties of sine and cosine to simplify the expression. Steps for simplification:

• First, express functions in terms of sine and cosine.
• Cancel out common factors from both the numerator and denominator. In this case, \( \sin x \) was cancelled out.
• Check your final expression and compare it to the given identity to confirm it's correct.

Using these steps makes it easier to see the relationships between components of the equation, leading to a better understanding and quicker solving of trigonometric problems in general. Simplifying expressions can also uncover patterns that might not be immediately obvious.