Problem 55 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$(1+\cos 2 \theta)(1-\cos 2 \theta)=\sin ^{2} 2 \theta$$

Step-by-Step Solution

Answer

The left-hand side simplifies to \(\sin^2 2\theta\), confirming the identity.

Step 1: Recognize the Identity

We begin with the left-hand side of the equation \((1+\cos 2\theta)(1-\cos 2\theta)\). This expression is structured similarly to the difference of squares, a common identity in algebra.

Step 2: Apply the Difference of Squares

The identity for the difference of squares is \((a-b)(a+b) = a^2 - b^2\). By comparing, we identify that \(a = 1\) and \(b = \cos 2\theta\). Substitute these into the formula: \((1)^2 - (\cos 2\theta)^2 = 1 - \cos^2 2\theta\).

Step 3: Use the Pythagorean Identity

Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Rearrange it to express \(\sin^2 x\) as \(\sin^2 x = 1 - \cos^2 x\). In our case, we replace \(x\) with \(2\theta\): \(\sin^2 2\theta = 1 - \cos^2 2\theta\).

Step 4: Verify the Final Equality

Substitute \(\sin^2 2\theta\) from Step 3 into the expression obtained in Step 2: \(1 - \cos^2 2\theta = \sin^2 2\theta\). Thus, the original left-hand expression is equal to the right-hand side, verifying the given identity.

Key Concepts

Difference of Squares

The difference of squares is a basic algebraic identity with significant application in solving equations. It is seen in expressions such as \((a+b)(a-b) = a^2 - b^2\). Here, the operation simplifies a product of two binomials.

In the problem at hand, we transform \((1 + \cos 2\theta)(1 - \cos 2\theta)\) into a more convenient form. This expression perfectly fits the difference of squares identity, where \(a = 1\) and \(b = \cos 2\theta\).

• This means the expression becomes \(1^2 - (\cos 2\theta)^2\), simplifying to \(1 - \cos^2 2\theta\).
• Recognizing such patterns can greatly simplify complex trigonometric setup.

Understanding and applying the difference of squares helps to bridge algebraic concepts with trigonometric identities. By simplifying through this identity, calculations become more manageable and less prone to error.

Pythagorean Identity

The Pythagorean identity, a cornerstone of trigonometry, states \(\sin^2 x + \cos^2 x = 1\). This identity emerges from the geometric relationship within a right triangle on the unit circle.

For our problem, we refashion it to express \(\sin^2 x\), which means \(\sin^2 x = 1 - \cos^2 x\). This step is crucial in transforming the expression resulting from applying the difference of squares.

• In our context, replace \(x\) with \(2\theta\) to get \(\sin^2 2\theta = 1 - \cos^2 2\theta\).
• By substituting this identity, we confirm that the left side of our original equation is indeed equal to the right side, solving the given identity.

This identity links angles and their trigonometric functions in a simple equation that holds true for any angle, making it versatile for many trigonometric transformations and simplifications.

Trigonometric Functions

Trigonometric functions are foundational to understanding the relationship between angles and sides in right triangles. In this context, we deal with \(\sin(2\theta)\) and \(\cos(2\theta)\), which are functions representing ratios derived from the unit circle.

These functions have key properties and identities that provide simplifications in equations, like the double angle formulas:

• \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
• \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)

In our specific trigonometric identity transformation, recognizing how these functions relate under different identities is essential. This knowledge allows one to manipulate expressions to verify identities effectively.

• These functions, derived from angles, help solve a broad array of trigonometric problems by offering various perspectives and forms of equations.

Gaining familiarity with these functions enhances problem-solving skills and deepens understanding of periodic phenomena in mathematics and applied fields like physics.