Problem 26 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(1 - \sin x\).

Step 1: Convert to Basic Trigonometric Functions

To simplify \[\frac{\cos x}{\sec x + \tan x},\]we start by converting the secant and tangent functions into sine and cosine functions using their identities. Recall:- \(\sec x = \frac{1}{\cos x}\)- \(\tan x = \frac{\sin x}{\cos x}\).So, replace the expression and rewrite it as:\[\frac{\cos x}{\frac{1}{\cos x} + \frac{\sin x}{\cos x}}.\]

Step 2: Find a Common Denominator

In the denominator, the terms are: \(\frac{1}{\cos x}\) and \(\frac{\sin x}{\cos x}\). They already have a common denominator of \(\cos x\). Thus, you can combine them:\[\frac{1 + \sin x}{\cos x}.\]

Step 3: Simplify the Expression

Now, rewrite the entire original expression \[\frac{\cos x}{\sec x + \tan x} \text{ as } \frac{\cos x}{\frac{1+\sin x}{\cos x}}.\]When dividing by a fraction, multiply by its reciprocal:\[\cos x \times \frac{\cos x}{1 + \sin x} = \frac{\cos^2 x}{1 + \sin x}.\]

Step 4: Apply Trigonometric Identity

Use the Pythagorean identity \(\cos^2 x = 1 - \sin^2 x\):\[\frac{1 - \sin^2 x}{1 + \sin x}.\]Notice that \(1 - \sin^2 x\) is the difference of squares, which can be factored:\[\frac{(1 - \sin x)(1 + \sin x)}{1 + \sin x}.\]

Step 5: Cancel Out Common Factors

In the expression:\[\frac{(1 - \sin x)(1 + \sin x)}{1 + \sin x},\]the terms \(1 + \sin x\) in the numerator and denominator cancel out, leaving:\[1 - \sin x.\]

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools that allow us to manipulate and simplify trigonometric expressions. These identities are formulae that relate the six basic trigonometric functions: sine (\( \sin \theta \)), cosine (\( \cos \theta \)), tangent (\( \tan \theta \)), cosecant (\( \csc \theta \)), secant (\( \sec \theta \)), and cotangent (\( \cot \theta \)). Understanding these relationships helps simplify otherwise complex expressions.

• The secant function is defined as the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \).
• The tangent function can be expressed in terms of sine and cosine: \( \tan x = \frac{\sin x}{\cos x} \).

Recognizing these identities allows for conversion of terms into the most basic form, often sine and cosine, which is a foundational step in simplifying expressions. Mastering trigonometric identities is essential in algebra, calculus, and beyond.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form using algebraic and trigonometric manipulations. When you're simplifying a trigonometric expression, it's often helpful to express all terms in a uniform manner; in this case, using sine and cosine.

In our example, \( \frac{\cos x}{\sec x+\tan x} \), we begin by rewriting the secant and tangent using their identities as mentioned in the previous section. We then aim to combine like terms or fractions by finding a common denominator. A shared denominator allows us to combine terms cohesively, simplifying the process.

• Find the common denominator for the terms in the expression.
• Transform the expression by multiplying and simplifying as required.
• Watch for opportunities to apply further identities or cancellation.

Simplifying makes the expression more manageable and often reveals hidden insights, like cancelling out terms or spotting equations that can be resolved quickly.

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry that stems from the Pythagorean Theorem. It states:\[ \sin^2 x + \cos^2 x = 1 \]This identity can be manipulated to solve and simplify various trigonometric equations. For example, one of its rearrangements is:\[ \cos^2 x = 1 - \sin^2 x \]
This is particularly useful in solving trigonometric expressions that involve \( \cos^2 x \) or \( \sin^2 x \).

In our original problem, this identity allowed us to replace \( \cos^2 x \) with \( 1 - \sin^2 x \), simplifying the expression further: \( \frac{1 - \sin^2 x}{1 + \sin x} \).

• Recognize expressions as potential applications of the Pythagorean Identity.
• Use the identity to transform and reduce the expression.
• Look for factorable forms such as differences of squares.

By applying the identity strategically, we can complete simplifications just like in the expression, ultimately reaching its simplest resolved form: \( 1 - \sin x \).