Problem 55 - Algebra and Trigonometry with Analytic Geometry

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 identity is verified; both sides equal \(\sin^2 2\theta\).

Step 1: Recognize the Expression Type

The left-hand side expression \((1 + \cos 2\theta)(1 - \cos 2\theta)\) is a product of two conjugates, which is a difference of squares formula.

Step 2: Apply the Difference of Squares Formula

Use the formula \(a^2 - b^2 = (a-b)(a+b)\) to transform the expression to \(1^2 - (\cos 2\theta)^2\). This simplifies to \(1 - \cos^2 2\theta\).

Step 3: Use a Trigonometric Identity to Simplify

Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). We can rearrange it to find \(\sin^2 x = 1 - \cos^2 x\). Thus, \(1 - \cos^2 2\theta = \sin^2 2\theta\).

Step 4: Verify the Identity

Now, we have the left-hand side expression \((1 - \cos^2 2\theta)\) transformed to \(\sin^2 2\theta\), which matches the right-hand side of the given equation.

Key Concepts

Difference of Squares

The concept of the "difference of squares" is a powerful algebraic tool that helps simplify products of conjugates. In mathematics, a conjugate refers to a binomial expression like \(a - b\) and \(a + b\). When multiplied, these conjugates result in a difference of squares. The general formula for this is \(a^2 - b^2 = (a-b)(a+b)\). This formula shows that the product of two conjugates can be simplified into a subtraction between two squares.

In the context of trigonometry, identifying expressions that fit this format is crucial. For instance, in the given identity \(1 + \cos 2\theta\) and \(1 - \cos 2\theta\), they form a conjugate pair. When these are multiplied, it matches the difference of squares formula, allowing us to write the expression as \((1)^2 - (\cos 2\theta)^2\). This becomes \(1 - \cos^2 2\theta\), simplifying the identity and setting the stage for further simplification using trigonometric identities.

Pythagorean Identities

Pythagorean identities are fundamental relationships in trigonometry that involve the square of sine and cosine functions. The most common Pythagorean identity is related to the unit circle, stating that for any angle \(x\), the square of the sine plus the square of the cosine equals one: \(\sin^2 x + \cos^2 x = 1\). This identity is a direct result of the Pythagorean theorem applied to the unit circle.

• Identity Form: \(\sin^2 x + \cos^2 x = 1\)
• Derived Form: \(\sin^2 x = 1 - \cos^2 x\)

Rearranging the identity allows us to express the square of sine in terms of the square of cosine, as used in the original exercise. When \(\sin^2 x\) is isolated, it gives us \(\sin^2 x = 1 - \cos^2 x\), which is critical when we need to simplify expressions like \(1 - \cos^2 2\theta\) into \(\sin^2 2\theta\). Understanding and applying Pythagorean identities help in verifying and transforming trigonometric identities effectively.

Cosine and Sine Functions

Cosine and sine are the backbone functions of trigonometry, representing fundamental periodic relationships. They are defined for any angle \(\theta\) and have values that oscillate between -1 and 1. The functions \(\cos 2\theta\) and \(\sin 2\theta\) specifically describe rotations and can be expressed in terms of the angle's double.

• **Cosine Function:** Represents the horizontal coordinate of an angle on the unit circle.
• **Sine Function:** Represents the vertical coordinate of an angle on the unit circle.

Both functions are interrelated through their identities and symmetries, such as in the Pythagorean identity. The cosine function, \(\cos 2\theta\), often appears within products that can be rewritten using trigonometric identities to simplify expressions, as shown in changing \(1 - \cos^2 2\theta\) to \(\sin^2 2\theta\). Understanding these functions and how they relate to one another is crucial for verifying and manipulating trigonometric identities.