Problem 62 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\sin x+\cos x}{\cos x}=1+\tan x $$

Step-by-Step Solution

Answer

The identity is verified as \( \frac{\sin x + \cos x}{\cos x} = 1 + \tan x \).

Step 1: Start with the Left Side

The left side of the equation is \( \frac{\sin x + \cos x}{\cos x} \). We will try to simplify this expression.

Step 2: Divide Each Term by cos(x)

Rewrite the expression by dividing each term inside the fraction by \( \cos x \):\[ \frac{\sin x}{\cos x} + \frac{\cos x}{\cos x} \].

Step 3: Simplify Each Term

The term \( \frac{\sin x}{\cos x} \) simplifies to \( \tan x \), and \( \frac{\cos x}{\cos x} \) simplifies to 1. Thus, the expression becomes \( \tan x + 1 \).

Step 4: Compare with the Right Side

The expression \( \tan x + 1 \) is exactly \( 1 + \tan x \), which matches the right side of the identity provided. Thus, the identity is verified.

Key Concepts

Verifying Trigonometric Identities

Verifying trigonometric identities is a process used to show that two sides of an equation are equivalent by using known trigonometric properties and identities. To approach these problems, start by choosing the most complex side of the equation. Working on this side can often lead to simplification that makes verifying the identity easier. Begin by writing down the more complex side and manipulate it using basic trigonometric identities such as Pythagorean identities, quotient identities, and co-function identities.

In the exercise provided, the left side of the equation was selected for simplification. The goal is to transform the left side into the same form as the right side of the equation. This involves using identities and making step-by-step transformations. By often breaking down each part of the expression, like dividing by a common function, it becomes straightforward to simplify towards the goal form. As seen in our solution, careful simplification confirmed that both sides of the equation were equivalent, verifying the identity.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing complex or long trigonometric equations into more manageable forms. This process uses known trigonometric identities so that equations can be solved or verified more easily. Key simplifications include changing expressions of sine and cosine into tangent or vice versa.

In the exercise, the given expression is \( \frac{\sin x + \cos x}{\cos x} \). To simplify it, each term in the numerator was divided by \( \cos x \). This action was informed by the quotient identity which defines tangent, allowing each part of the fraction to be simplified individually:

• \( \frac{\sin x}{\cos x} \) simplifies to \( \tan x \)
• \( \frac{\cos x}{\cos x} \) simplifies to 1

By breaking down the expression correctly, these simplifications not only verified the identity but also highlighted effective techniques for rewriting trigonometric expressions. Understanding how to perform these steps is crucial in both verifying identities and solving trigonometric equations in general.

Tangent Function

The tangent function, often represented as \( \tan x \), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function, which can be written as \( \tan x = \frac{\sin x}{\cos x} \). The tangent function is periodic and has a period of \( \pi \), which means it repeats its pattern every \( \pi \) units.

In the provided problem, understanding the tangent function was key in simplifying the expression. Recognizing that dividing sine by cosine results in tangent, allowed the simplification of \( \frac{\sin x}{\cos x} \) to \( \tan x \). This transformation is not only crucial within the context of the exercise but also in many facets of trigonometry where simplification can lead to easier comprehensible forms.

• This property of tangent was crucial for the verification step
• It is also frequently used in calculus and real-world applications like engineering

Awareness of this function and its properties helps in solving and simplifying a wide variety of mathematical problems.