Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\cos^2 x}{1 + \sin x} \).

Step 1: Identify Reciprocal Identities

Recall that the secant and tangent functions can be rewritten in terms of sine and cosine. Specifically, \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these into the expression.

Step 2: Substitute Identities into Expression

Substitute the reciprocal identities into the expression: \( \frac{\cos x}{\sec x + \tan x} = \frac{\cos x}{\frac{1}{\cos x} + \frac{\sin x}{\cos x}} \). The denominator now needs to be simplified by combining into a single fraction.

Step 3: Combine Denominators

In the denominator, convert to a common denominator which is \( \cos x \): \( \frac{1}{\cos x} + \frac{\sin x}{\cos x} = \frac{1 + \sin x}{\cos x} \). Now the expression is \( \frac{\cos x}{\frac{1 + \sin x}{\cos x}} \).

Step 4: Simplify the Expression

The expression is now a complex fraction. Simplify it by multiplying the numerator by the reciprocal of the denominator: \( \cos x \times \frac{\cos x}{1 + \sin x} \). This simplifies to \( \frac{\cos^2 x}{1 + \sin x} \).

Step 5: Use Pythagorean Identity

Utilize the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \) to further simplify if needed in specific contexts. However, this formulation does not reduce further in simple form relative to \( 1 + \sin x \) so the expression is simplified as \( \frac{\cos^2 x}{1 + \sin x} \).

Key Concepts

Reciprocal Identities

Trigonometric functions have reciprocal identities that allow them to be rewritten in different ways. Understanding these identities is essential when simplifying trigonometric expressions.

Common reciprocal identities include:

• Secant (\( \sec x \)) is the reciprocal of cosine (\( \cos x \)), so:\[ \sec x = \frac{1}{\cos x} \]
• Cosecant (\( \csc x \)) is the reciprocal of sine (\( \sin x \)), so:\[ \csc x = \frac{1}{\sin x} \]
• Cotangent (\( \cot x \)) is the reciprocal of tangent (\( \tan x \)), so:\[ \cot x = \frac{1}{\tan x} \]

These identities are used to convert more complex expressions into simpler forms by changing them from fractions to trigonometric functions or vice versa. In our given exercise, replacing \( \sec x \) and \( \tan x \) with their equivalent reciprocal forms allows us to simplify the overall expression easily.

Pythagorean Identities

Pythagorean identities are derived from the Pythagorean theorem and are vital in simplifying trigonometric expressions further.

The most basic Pythagorean identity is:\[ \sin^2 x + \cos^2 x = 1 \]

From this, you can derive other forms by rearrangement:

• \[ 1 - \sin^2 x = \cos^2 x \]
• \[ 1 - \cos^2 x = \sin^2 x \]
• \[ \sec^2 x = 1 + \tan^2 x \]
• \[ \csc^2 x = 1 + \cot^2 x \]

These identities are particularly helpful when expressions include squares of sine or cosine. In the exercise, we advised using \( \cos^2 x = 1 - \sin^2 x \) to examine if further simplification is possible. However, sometimes, simplification stops at a certain step when no more reduction is plausible.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic principles to reduce the expression to its simplest form.

Here are a few strategies to simplify expressions effectively:

• Identify all identities that can transform parts of the expression into simpler terms.
• If fractions are involved, find a common denominator to combine terms.
• Convert complex fractions into simpler forms by multiplying the numerator by the reciprocal of the denominator.
• Check for opportunities to apply Pythagorean identities, especially when the expression involves squared terms.

In the given problem, we started with substituting reciprocal identities to convert secant and tangent into terms of sine and cosine. After simplifying the complex fraction, we multiplied by the reciprocal of the denominator, leading to a simpler form using \( \frac{\cos^2 x}{1 + \sin x} \). Approaching problems systematically ensures clarity and correctness, especially in more complex scenarios.