Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \sin ^{4} \theta+2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=1 $$

Step-by-Step Solution

Answer

The identity is verified since \((\sin^2 \theta + \cos^2 \theta)^2 = 1\).

Step 1: Recognize the Pythagorean Identity

Recall the Pythagorean identity, which states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This foundational trigonometric identity will be useful in simplifying the expression.

Step 2: Rewrite Existing Terms

Observe that the given expression \( \sin^4 \theta + 2 \sin^2 \theta \cos^2 \theta + \cos^4 \theta \) can be grouped as \( (\sin^2 \theta)^2 + 2(\sin^2 \theta)(\cos^2 \theta) + (\cos^2 \theta)^2 \). This form looks similar to the expansion of the binomial theorem \( (a + b)^2 \).

Step 3: Expand Using the Perfect Square

Notice that \( a^2 + 2ab + b^2 = (a + b)^2 \), thus this implies \[ (\sin^2 \theta + \cos^2 \theta)^2. \]

Step 4: Apply the Pythagorean Identity

Substitute the Pythagorean identity into the expression: \( (\sin^2 \theta + \cos^2 \theta) = 1 \). Therefore, \[ (\sin^2 \theta + \cos^2 \theta)^2 = 1^2. \]

Step 5: Simplify the Expression

Simplify \( 1^2 \) to 1. Hence, the identity holds as \( \sin^4 \theta + 2 \sin^2 \theta \cos^2 \theta + \cos^4 \theta = 1 \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a key concept in trigonometry. It states that for any angle \( \theta \), the relation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is derived from the Pythagorean theorem, which relates the sides of a right-angled triangle. In the context of the unit circle, it represents fundamental properties of sine and cosine functions. The identity helps in simplifying and transforming expressions involving sine and cosine.

This identity is especially useful when verifying complex trigonometric identities or simplifying expressions, as it allows us to replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \) (and vice versa). This substitution can reveal simpler forms of expressions or make them easier to manipulate. In our exercise, the Pythagorean identity is the core transformation that leads to the simplification of the given expression.

Simplifying Expressions

Simplifying expressions in mathematics involves reducing them to their simplest form. This means taking complex or lengthy expressions and rewriting them so they are easier to work with or understand. Trigonometry provides a number of identities, like the Pythagorean identity, to aid in this task.

In the provided exercise, the aim is to transform the complicated expression \( \sin^4 \theta + 2 \sin^2 \theta \cos^2 \theta + \cos^4 \theta \) into something recognizably simpler. Achieving this involves leveraging algebraic manipulations, like recognizing patterns and applying identities.

• Firstly, notice how the expression resembles the expansion of the binomial theorem, which simplifies its complexity.
• Secondly, we apply the Pythagorean identity to further reduce the expression.

By matching known identities and patterns, expressions that initially seem daunting become manageably simple. A systematic approach to simplification not only provides the correct answer but also deepens our understanding of mathematical properties and relationships.

Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions raised to a power. It states that any binomial, an expression like \( (a + b)^n \), can be expanded into a sum involving powers of \( a \) and \( b \). For the expression \( (a + b)^2 \), the expansion is \( a^2 + 2ab + b^2 \). This relates directly to our exercise, where the expression \( \sin^4 \theta + 2 \sin^2 \theta \cos^2 \theta + \cos^4 \theta \) mirrors the result of squaring a binomial.

Understanding the binomial theorem:

• It provides a systematic way of expanding powers of sums.
• It involves coefficients, known as binomial coefficients, which are derived from Pascal's triangle.
• Recognizing patterns from the binomial theorem facilitates quicker simplification of terms.

In this case, identifying that the complex expression is an exact match to the expansion \( (\sin^2 \theta + \cos^2 \theta)^2 \), allows us to apply the binomial theorem effectively. By converting it into a squared binomial form, it becomes straightforward to apply the Pythagorean identity and verify the original identity proposed in the exercise.