Problem 66 - Precalculus with Limits

Question

In Exercises 59 - 70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. \( 1 - 2 \cos^2 x + \cos^4 x \)

Step-by-Step Solution

Answer

The simplified expression of \(1 - 2 \cos^2 x + \cos^4 x\) is \(\sin^4 x\).

Step 1: Factor the expression

As the given expression \(1 - 2 \cos^2 x + \cos^4 x\) is a quadratic formula in \(\cos^2 x\), it can be rewritten as \( (\cos^2 x -1)^2\)

Step 2: Apply trigonometric identity

According to the Pythagorean trigonometric identity, one can replace \(\cos^2 x\) with \(1 - \sin^2 x\). So we get \((1 - \sin^2 x - 1)^2 = -\sin^4 x\)

Step 3: Simplify the expression

An even power to a negative number will result in a positive number. Therefore, \(-\sin^4 x\) simplifies to \(\sin^4 x\).

Key Concepts

Fundamental Trigonometric Identities

Trigonometry is rife with relationships between the ratios of angles and sides of triangles. At the core of these relationships are the fundamental trigonometric identities, which are equations involving trigonometric functions that are true for every value of the variable involved.

These identities are crucial for simplifying trigonometric expressions and solving trigonometric equations. Key fundamental identities include the reciprocal identities, such as \( \frac{1}{\sin x} = \csc x \) and \( \frac{1}{\cos x} = \sec x \), and the quotient identities like \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

Understanding these identities allows students to rewrite trigonometric expressions in different forms. For instance, an expression involving \( \sin x \) and \( \cos x \) can often be transformed into a single function, like \( \tan x \) or \( \cot x \) using these identities.

Pythagorean Trigonometric Identity

One of the most powerful and frequently used identities in trigonometry is the Pythagorean trigonometric identity. It stems from the Pythagorean theorem, which links the sides of a right-angled triangle. The identity itself expresses a relationship between the squares of the sine and cosine of an angle.

The fundamental form of the Pythagorean identity is \( \sin^2 x + \cos^2 x = 1 \). This can be manipulated to express either the sine or cosine function in terms of the other: \( \sin^2 x = 1 - \cos^2 x \) and \( \cos^2 x = 1 - \sin^2 x \).

In the context of the given exercise, the Pythagorean identity is used to replace \( \cos^2 x \) with \( 1 - \sin^2 x \) in order to simplify the expression further. This substitution is a powerful technique in factoring and simplifying trigonometric expressions, making the identity indispensable for students to master.

Simplifying Trigonometric Expressions

The process of simplifying trigonometric expressions often involves factoring, the use of fundamental identities, and algebraic manipulation. The goal is to reduce the expression to its simplest form or to a specific requested form.

To simplify a complex trigonometric expression, students typically follow these steps:

• Recognize the form of the expression and factor it if possible.
• Apply fundamental and Pythagorean trigonometric identities to substitute and eliminate variables.
• Combine like terms and simplify the exponents.
• Check the result against known identities to see if further simplification is possible.

In our exercise, the expression begins in a quadratic form and is then factored. The fundamental identity involving the Pythagorean trigonometric identity is used to substitute for \( \cos^2 x \), and from there, algebraic manipulation brings it down to the simplest form. By mastering these steps, students can approach trigonometric simplification with confidence.