Problem 16 - 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: Substitute Trigonometric Identities

First, we substitute trigonometric identities where possible. Recall that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substituting these into the expression gives:\[\frac{\frac{1}{\cos x} - \cos x}{\frac{\sin x}{\cos x}}\]

Step 2: Simplify the Numerator

Combine the terms in the numerator under a common denominator:\[\frac{1}{\cos x} - \cos x = \frac{1 - \cos^2 x}{\cos x}\]Notice that \(1 - \cos^2 x \) is equivalent to \( \sin^2 x \) using the Pythagorean identity.

Step 3: Combine into One Fraction

The expression now becomes:\[\frac{\sin^2 x}{\cos x} \div \frac{\sin x}{\cos x}\]Dividing by a fraction is equivalent to multiplying by its reciprocal, so this becomes:\[\frac{\sin^2 x}{\cos x} \times \frac{\cos x}{\sin x}\]

Step 4: Simplify the Expression

Cancel the common terms \( \cos x \) from the numerator and denominator:\[\frac{\sin^2 x}{\sin x}\]Further simplify by canceling \( \sin x \):\[ \sin x\]

Step 5: Conclusion

The simplified form of the original trigonometric expression is \( \sin x \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations in trigonometry that relate the functions such as sine, cosine, tangent, and their reciprocals. Understanding and using these identities can simplify complex trigonometric expressions more easily.

• One useful identity is the reciprocal identity which states that the secant of an angle is the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \).
• Similarly, the tangent of an angle can be expressed in terms of sine and cosine: \( \tan x = \frac{\sin x}{\cos x} \).

By substituting these identities into expressions, you can transform and simplify them into more familiar or more manageable forms. In this way, working with trigonometric identities allows for flexibility and creativity in solving trigonometric problems. By comprehending these relationships, you develop a toolkit for simplifying expressions or solving equations effectively.

Pythagorean Identity

The Pythagorean identity is one of the most central trigonometric identities. It stems from the Pythagorean theorem and connects sine and cosine thereby providing a powerful tool for expression simplification. The identity states:
\[ \sin^2 x + \cos^2 x = 1 \]
This can be rearranged to express one function in terms of the other, such as \( \sin^2 x = 1 - \cos^2 x \) or \( \cos^2 x = 1 - \sin^2 x \).
These equivalent forms are particularly useful in simplification, as evidenced in the original exercise where \( 1-\cos^2 x \) is replaced by \( \sin^2 x \). This change helps in reducing the complexity of the trigonometric expression to something simpler and more recognizable. The ability to seamlessly switch between these equivalent forms is crucial in converting a tangled expression into a solvable one.

Expression Simplification

Expression simplification in trigonometry involves using known identities to make an expression easier to interpret or solve. The goal is to transform complex expressions into simpler forms that are more straightforward to work with.
In the original exercise, the expression \( \frac{\sec x - \cos x}{\tan x} \) was simplified through a series of steps:

• Substituting trigonometric identities like \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \) to transform the complex expression.
• Combining the terms under a common denominator in the numerator to reach \( \frac{1 - \cos^2 x}{\cos x} \), then applying the Pythagorean identity to simplify this to \( \frac{\sin^2 x}{\cos x} \).
• Performing operations like division by a fraction (which is the same as multiplying by its reciprocal) to further simplify the expression.
• Finally, canceling common terms in the numerator and the denominator to reach the result \( \sin x \).

Each of these steps relies on a foundational understanding of how the various trigonometric functions relate to one another. Once simplified, the expression is easier to use, whether in solving equations, graphing, or further manipulation.