Problem 39 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \tan(-x) \cos x \)

Step-by-Step Solution

Answer

The simplified expression for \( \tan(-x) \cos x \) is \( -\sin x \).

Step 1: Apply the tan identity

Replace \( \tan(-x) \) with \( -\tan x \), using the fact that \( \tan(-x) = - \tan x \). So the expression becomes \( -\tan x \cos x \).

Step 2: Replace the tan expression

Utilize the identity \( \tan x = \sin x / \cos x \). This turns the expression into \( -( \sin x / \cos x ) \cos x \).

Step 3: Simplification

Simplify the expression by multiplying the terms: \( \sin x / \cos x * \cos x \) simplifies to \( -\sin x \).

Key Concepts

The Tangent Identity

The tangent identity is a key tool in trigonometry, allowing us to transform and simplify expressions involving the tangent function.
In trigonometry, the tangent function has an odd identity, which states:

• \( \tan(-x) = -\tan x \)

This means that the tangent of a negative angle is the negative of the tangent of the angle itself.
This identity can be very useful when you're trying to simplify expressions, as it allows you to replace negative angles with positive ones, thereby making the expression easier to work with.
In our example, applying the tangent identity transforms \( \tan(-x) \cos x \) into \( -\tan x \cos x \), making the expression simpler to handle in subsequent steps.

Simplifying Expressions with Trigonometry

Simplifying expressions involves reducing complex expressions into simpler forms without changing their values.
When dealing with trigonometric functions like tangent, sine, and cosine, it often involves using identities to rewrite parts of the expression.

• Replace \( \tan x \) with \( \frac{\sin x}{\cos x} \)

By replacing \( \tan x \) with \( \frac{\sin x}{\cos x} \), we express tangent in terms of sine and cosine, which can make the expression easier to simplify.
This step turns our expression from \( -\tan x \cos x \) into \( -\left( \frac{\sin x}{\cos x} \right) \cos x \).
Recognizing these transformations allows us to break down the problem and work towards a simpler solution.

Trigonometric Simplification Techniques

Trigonometric simplification is all about reducing expressions using trigonometric identities.
This often involves canceling terms or combining like expressions:

• Multiply fractions: \( \left( \frac{\sin x}{\cos x} \right) \cos x \)
• Cancel out common terms: The \( \cos x \) in the numerator and denominator cancels out

You're left with \( -\sin x \), which is much simpler than the original expression.
Being able to see these cancellations and transformations is key to mastering trigonometric simplification.
Always aim to express trigonometric identities in their simplest forms to make calculations quicker and more intuitive.