Problem 63 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\sin x-\cos x}{\tan x}=\frac{\tan x}{\sin x+\cos x}$$

Step-by-Step Solution

Answer

Based on the step by step solution, determine if the given equation is an identity or not: Answer: The given equation is not an identity.

Step 1: Convert the expressions to sine and cosine functions

Given equation: $$\frac{\sin x-\cos x}{\tan x}=\frac{\tan x}{\sin x+\cos x}$$ To work with the main trigonometric functions (sine and cosine), we first rewrite the given equation in terms of sine and cosine functions: $$\frac{\sin x-\cos x}{\frac{\sin x}{\cos x}}=\frac{\frac{\sin x}{\cos x}}{\sin x+\cos x}$$

Step 2: Simplify both sides of the equation

Next, we simplify both sides of the equation by multiplying the denominators by their respective numerators. Left side: $$\frac{\sin x-\cos x}{\frac{\sin x}{\cos x}} \cdot \frac{\cos x}{\cos x}=\frac{(\sin x -\cos x)(\cos x)}{\sin x}$$ Right side: $$\frac{\frac{\sin x}{\cos x}}{\sin x+\cos x} \cdot \frac{\cos x}{\cos x}=\frac{\sin x}{\sin x + \cos x}$$

Step 3: Prove the equation

Now, we need to show that the left side of the equation is equal to the right side of the equation: Left side: $$\frac{(\sin x -\cos x)(\cos x)}{\sin x}$$ Right side: $$\frac{\sin x}{\sin x + \cos x}$$ In order to compare the two sides, we rewrite the left side by distributing the cosine: Left side: $$\frac{\sin x\cos x -\cos^2 x}{\sin x}$$ Now, we can compare the left and right sides directly: $$\frac{\sin x\cos x -\cos^2 x}{\sin x} \neq \frac{\sin x}{\sin x + \cos x}$$ Since the left and right sides of the equation are not equal, the given equation is not an identity.

Key Concepts

Sine and Cosine Functions

Understanding sine and cosine functions is fundamental in trigonometry. They are the building blocks used to describe periodic phenomena, such as waves.

• The sine function, denoted as \( \sin x \), gives the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function, denoted as \( \cos x \), provides the ratio of the adjacent side to the hypotenuse.

Both these functions are crucial when expressing other trigonometric functions. For instance, the tangent function, \( \tan x \), can be written as \( \frac{\sin x}{\cos x} \). This relationship helps in simplifying expressions, which we will explore further in this article.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves rewriting them using basic trigonometric identities. This process makes solving equations or proving identities more manageable. Let's break it down into straightforward steps:

• Convert compound trigonometric functions into sine and cosine. This conversion aids in uniformity, making it easier to equate or simplify the equations.
• Apply basic identities such as \( \tan x = \frac{\sin x}{\cos x} \) whenever possible to streamline expressions.

For example, in our original equation \( \frac{\sin x - \cos x}{\tan x} = \frac{\tan x}{\sin x + \cos x} \), we converted \( \tan x \) into \( \frac{\sin x}{\cos x} \). This re-expression allows us to approach the equation more uniformly. Simplification typically involves multiplying expressions by appropriate terms to eliminate complex fractions. This technique helps clear up clutter and gives us a clearer view of the relationship between terms.

Proving Non-Identities

Unlike identities, which hold true for all values of a variable, non-identities do not. The goal in proving or disproving an identity is to show whether both sides of an equation are identical through simplification and comparison. Here's how we do it:

• First, simplify each side of the equation as much as possible.
• Compare the simplified forms of both sides. If they are equal, it proves the equation is an identity.

In the example equation, after simplification and comparison, we found inequivalence. This discovery confirmed that the equation \( \frac{\sin x \cos x - \cos^2 x}{\sin x} eq \frac{\sin x}{\sin x + \cos x} \), illustrating it's not an identity. Such an analysis suggests the expressions differ for at least some angle values of \( x \), reinforcing that the left doesn't always equal the right across all conditions. Recognizing non-identities helps in understanding the boundaries of trigonometric relationships and the specific behavior of functions.