Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides are equal.

Step 1: Expand both sides of the identity

We start by examining the left side of the equation: \[\cos^4 2\theta + \sin^2 2\theta\]Rewriting \(\cos^4 2\theta\) as \((\cos^2 2\theta)^2\), the expression becomes:\[(\cos^2 2\theta)^2 + \sin^2 2\theta\]Now for the right side:\[\cos^2 2\theta + (\sin^2 2\theta)^2\]Both sides seem distinct at this stage, indicating detailed steps are necessary to verify equality.

Step 2: Apply Pythagorean identity

Recall the Pythagorean identity: \[\cos^2 x + \sin^2 x = 1\]Substituting \(x = 2\theta\), we have:\[\cos^2 2\theta = 1 - \sin^2 2\theta\] and \[\sin^2 2\theta = 1 - \cos^2 2\theta\]These identities can aid in manipulating the given identity.

Step 3: Simplify the expressions

Replace \(\cos^2 2\theta\) in the left side equation with \(1 - \sin^2 2\theta\):\[(1 - \sin^2 2\theta)^2 + \sin^2 2\theta\]Expand \((1 - \sin^2 2\theta)^2\):\[1 - 2\sin^2 2\theta + (\sin^2 2\theta)^2 + \sin^2 2\theta\]Combine like terms, resulting in:\[1 - \sin^2 2\theta + (\sin^2 2\theta)^2\]This is equal to the expression on the right side of the identity.

Step 4: Verify both sides are equivalent

From Step 3, we showed that the simplified left side \[1 - \sin^2 2\theta + (\sin^2 2\theta)^2\]is indeed equal to the right side of the original identity, which is:\[\cos^2 2\theta + (\sin^2 2\theta)^2\]This confirms that both expressions are identical, verifying the original identity.

Key Concepts

Pythagorean Identity

Understanding trigonometric identities is crucial for solving many mathematical problems. One of the most fundamental identities is the Pythagorean Identity. Simply put, the Pythagorean Identity states: \[\cos^2 x + \sin^2 x = 1\] This equation is similar to the famous Pythagorean theorem in geometry, but instead of working with sides of a right triangle, we use trigonometric functions. The identity holds for any angle \(x\), describing the relationship between the sine and cosine of that angle.
In verifying trigonometric identities like the one given in the exercise, the Pythagorean Identity is often used to manipulate expressions into a simpler or more recognizable form. Understanding and recognizing how this identity can be applied is key to moving forward in trigonometric simplifications and proofs.

Trigonometrical Simplification

Trigonometrical simplification involves breaking down complex trigonometric expressions into simpler forms. This process is crucial for solving equations or verifying identities. Let's break down how this applies in our example:

• First, re-write powers of trigonometric functions using squares, such as expressing \(\cos^4 2\theta\) as \((\cos^2 2\theta)^2\).
• Utilize trigonometric identities, like the Pythagorean Identity, to substitute expressions such as \(\cos^2 2\theta\) with \(1 - \sin^2 2\theta\).

These steps help break down complex expressions, paving the way for easier calculation or comparison. Trigonometrical simplification is an acquired skill that develops with practice, helping in turning lengthy expressions into shorter, more manageable forms. This is a critical step in verifying that both sides of the given identity are equivalent.

Algebraic Manipulation

Algebraic manipulation lies at the heart of solving and verifying trigonometric identities. By carefully expanding, factoring, and simplifying algebraic expressions, we can prove that different looking expressions are indeed equal.
In the exercise, after applying the Pythagorean Identity, we moved forward with expanding and simplifying the expression. Here's a quick breakdown:

• Replace \((1 - \sin^2 2\theta)^2\) with its expanded form: \(1 - 2\sin^2 2\theta + (\sin^2 2\theta)^2\).
• Combine like terms, ensuring that each expression is as simplified as possible.

Algebraic manipulation requires precision as it's easy to make small mistakes that can propagate throughout your calculations. Always double-check each step, especially when expanding squares or simplifying terms. By mastering these manipulations, you enhance your trigonometric toolset, making complex identities more approachable and solvable.