Problem 18 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{\sec x-\cos x}{\tan x} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sin x \).

Step 1: Rewrite Secant and Tangent in Terms of Sine and Cosine

The first step is to express both \( \sec x \) and \( \tan x \) in terms of sine and cosine. Recall that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these into the expression:\[\frac{\frac{1}{\cos x} - \cos x}{\frac{\sin x}{\cos x}}\]

Step 2: Simplify the Numerator

Now simplify the numerator by combining the terms. The numerator becomes:\[\frac{1}{\cos x} - \cos x = \frac{1 - \cos^2 x}{\cos x}\]

Step 3: Use Trigonometric Identity

Recognize that \( 1 - \cos^2 x \) is a trigonometric identity for \( \sin^2 x \). Replace it in the expression:\[\frac{\sin^2 x}{\cos x}\]

Step 4: Divide by the Denominator

Now divide by the denominator, \( \frac{\sin x}{\cos x} \), which is actually multiplying by the reciprocal:\[\frac{\sin^2 x}{\cos x} \times \frac{\cos x}{\sin x} = \sin x\]

Step 5: Conclusion: Simplified Expression

After simplification, the entire expression reduces to \( \sin x \).

Key Concepts

Secant Function

The secant function, often denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is quite important for solving various trigonometric expressions and equations. Understanding how \( \sec x \) relates to the other trigonometric functions is key to simplifying expressions effectively.

• By definition, the secant function is the reciprocal of the cosine function. This means you can express it as \( \sec x = \frac{1}{\cos x} \).
• Knowing this reciprocal identity can help when you need to rewrite complex trigonometric expressions in simpler terms.

The step-by-step solution to the original exercise takes advantage of this definition. By substituting \( \sec x \) with \( \frac{1}{\cos x} \), the complex expression becomes more manageable. This changes the original fraction into a form that can more easily be simplified using other identities, setting the stage for the next steps.

Tangent Function

The tangent function, represented as \( \tan x \), is another primary trigonometric function. It's essential to understand its relationship with sine and cosine to simplify expressions efficiently.

• Tangent is the ratio of the sine function to the cosine function, so we have \( \tan x = \frac{\sin x}{\cos x} \).
• This relationship allows you to substitute \( \tan x \) with its equivalent in terms of \( \sin x \) and \( \cos x \), facilitating the combination and simplification of terms.

In the original solution, rewriting \( \tan x \) as \( \frac{\sin x}{\cos x} \) helps us align the expression with the already modified secant function. This gives us a unified language of sine and cosine, which is easier to simplify together.

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the basic building blocks of trigonometry. Understanding its properties and identities is crucial for trigonometric simplification.

• Cosine represents the adjacent side over the hypotenuse in a right triangle and is written as \( \cos x \).
• Many trigonometric identities involve \( \cos x \), including the famous Pythagorean identity: \( 1 - \cos^2 x = \sin^2 x \).

This identity plays a pivotal role in our simplification task. When we reached the term \( \frac{1 - \cos^2 x}{\cos x} \) in the solved expression, recognizing and substituting with \( \sin^2 x \) was the key step in simplifying further. Such identities make trigonometric expressions significantly more straightforward to handle.

Simplification Steps

Breaking down trigonometric expressions into their base components can vastly simplify the problem. The goal is to express complex functions in terms of sine and cosine and use known identities for further simplification. Here is a quick summary of the simplification approach applied in the exercise:

• Step 1: Substitute \( \sec x \) and \( \tan x \) using their definitions in terms of sine and cosine. This method standardizes the expression to \( \frac{\frac{1}{\cos x} - \cos x}{\frac{\sin x}{\cos x}} \).
• Step 2: Simplify the expression by combining like terms in the numerator, leading to \( \frac{1 - \cos^2 x}{\cos x} \).
• Step 3: Apply the Pythagorean identity \( 1 - \cos^2 x = \sin^2 x \) to simplify further.
• Step 4: Multiply the resulting expression \( \frac{\sin^2 x}{\cos x} \) by the reciprocal of the denominator to finally arrive at \( \sin x \).

Each step systematically reduces the complexity of the expression, demonstrating the power of strategic substitutions and the application of trigonometric identities.