Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and use the fundamental identities to simplify. $$\tan x+\frac{\cos x}{1+\sin x}$$

Step-by-Step Solution

Answer

The simplified expression is \(1 / \cos x\).

Step 1: Substitute tan with sin/cos

Replace \(\tan x\) with \(\sin x / \cos x\). The expression now is \(\sin x / \cos x + \cos x / (1 + \sin x)\)

Step 2: Find the common denominator

The common denominator of the two fractions is \( \cos x (1 + \sin x)\). We transform the fractions to have this common denominator: \((\sin x (1 + \sin x) + \cos^2 x) / (\cos x (1 + \sin x))\)

Step 3: Simplify the numerator

Expand and simplify the numerator. We now have \((\sin x + \sin^2 x + \cos^2 x) / (\cos x(1 + \sin x))\). But since we know that \( \sin^2 x + \cos^2 x = 1 \), the expression is simplified to \( (1 + \sin x) / (\cos x(1 + \sin x))\)

Step 4: Cancel out common terms

Cancel out the \(1 + \sin x\) in the numerator and denominator. We're then left with the final simplified expression: \( 1 / \cos x \)

Key Concepts

Fundamental Trigonometric Identities

Understanding fundamental trigonometric identities is crucial for simplifying trigonometric expressions. These identities represent the relationships between the trigonometric functions (sine, cosine, tangent, etc.) and are used to transform complex expressions into simpler ones.

Some of the most commonly used fundamental identities are:

• The Pythagorean Identity: \[ \(\sin^2x + \cos^2x = 1\) \]
• The Reciprocal Identities: \[ \frac{1}{\sin x} = \csc x\], \[\frac{1}{\cos x} = \sec x\], \[\frac{1}{\tan x} = \cot x\]
• The Quotient Identity: \[ \tan x = \frac{\sin x}{\cos x}\]

By using these identities, one can simplify an expression like \(\tan x + \frac{\cos x}{1 + \sin x}\) into a much simpler form. In the given exercise, the Pythagorean Identity allows us to turn \(\sin^2 x + \cos^2 x\) into 1, leading to significant simplification of the expression.

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate radicals or complex expressions from the denominators of fractions. It involves multiplying the numerator and the denominator by an appropriate expression that will make the denominator rational. This process is valuable when dealing with trigonometric identities where the goal is to have a more straightforward expression.

In the context of trigonometry, this can mean adjusting expressions involving sine and cosine to remove roots or to combine several terms into a single term. For instance, in the exercise \(\tan x + \frac{\cos x}{1 + \sin x}\), you would multiply the numerator and denominator by the conjugate of the denominator, which in this case is \(1 - \sin x\), to eliminate the complex denominator. However, in our specific problem, we notice that by using trigonometric identities, we can simplify the expression without rationalizing.

Trigonometric Addition and Subtraction

Trigonometric addition and subtraction involve combining or separating trigonometric functions within an equation. Mastering this concept helps in solving trigonometric equations and simplifying expressions.

Using trigonometric identities is essential when adding or subtracting terms. For the addition of fractions, as seen in the expression, a common denominator must be found. In our exercise, we combined \(\sin x / \cos x\) with \(\cos x / (1 + \sin x)\) by finding a common denominator \(\cos x(1 + \sin x)\). This step is critical before simplifying the numerator.

Subtracting works similarly, where trigonometric identities can help to rewrite the functions in a form that makes subtraction straightforward. By mastering these techniques, one can efficiently tackle a broad range of trigonometric problems, as seen in the solution to the exercise given.