Problem 65 - 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. \( \tan^4 x + 2 \tan^2 x + 1 \)

Step-by-Step Solution

Answer

The simplified form of the expression is \( sec^4x \).

Step 1: Identify The Pattern

Notice that the given expression has the form \( a^2 + 2ab + b^2 \). This is the pattern of a perfect square trinomial and it can be simplified as \((a + b)^2 \). In this case, \( a = tan^2x \) and \( b = 1 \).

Step 2: Implement The Factorisation

Substitute \( a \) and \( b \) into the pattern. The expression becomes \( (a + b)^2= (\tan^2x + 1)^2\).

Step 3: Use Trigonometric Identity

Recall the Pythagorean identity, where \( \tan^2x + 1 = sec^2x \). Substitute this identity into the expression. The simplified expression is \( (sec^2x)^2\).

Step 4: Simplify The Expression

Simplify \( (sec^2x)^2 \) to get \( sec^4x \).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of algebraic expression. It follows a specific pattern, which makes it easier to factor and simplify. The pattern is usually expressed as

• \(a^2 + 2ab + b^2\), which simplifies to \((a + b)^2\),
• or \(a^2 - 2ab + b^2\), which simplifies to \((a - b)^2\).

Identifying this pattern in expressions allows us to factor them quickly and efficiently.

In the given exercise, the expression \(\tan^4 x + 2 \tan^2 x + 1\) matches this pattern with \(a = \tan^2 x\) and \(b = 1\). The result is \((\tan^2 x + 1)^2\). Recognizing perfect square trinomials helps in streamlining the simplification process, reducing complex expressions into something more manageable.

Pythagorean Identity

Trigonometric identities are powerful tools in simplifying expressions. The Pythagorean identity is among the fundamental ones in trigonometry. The specific identity relevant to our exercise is:

• \(\tan^2 x + 1 = \sec^2 x\).

This identity is derived from the Pythagorean theorem and relates tangent and secant functions. By substituting this identity into our expression, \((\tan^2 x + 1)^2\) simplifies to \((\sec^2 x)^2\).

Understanding the Pythagorean identities enables easy conversion between different trigonometric functions, paving the way for further simplification. It's important to familiarize oneself with these identities as they are commonly used in various mathematical and engineering fields.

Factorization

Factorization involves breaking down a complex expression into simpler multiples that, when multiplied together, yield the original expression. This process is essential in solving equations, simplifying expressions, and performing integrations in calculus.

In the context of our exercise, factorization allowed us to identify and utilize the perfect square trinomial pattern, transforming \(\tan^4 x + 2 \tan^2 x + 1\) into \((\tan^2 x + 1)^2\). By then applying the Pythagorean identity, the expression further simplifies to \(\sec^4 x\).

The ability to factor expressions correctly is crucial in algebra. This not only simplifies the problem-solving process but also provides a clearer understanding of the relationships between different algebraic and trigonometric expressions.