Problem 56 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

Identity is verified: both sides simplify to \(\cos 4\theta\).

Step 1: Recognize the Trigonometric Identity

The identity provided in the exercise resembles a well-known trigonometric identity: the double angle formula for cosine, which is given by \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). We can apply this identity to the left-hand side expression \(\cos^2 2\theta - \sin^2 2\theta\).

Step 2: Apply the Double Angle Formula

Apply the double-angle formula for cosine to the expression \(\cos^2 2\theta - \sin^2 2\theta\), which states that \(\cos 2\theta = \cos^2\theta - \sin^2\theta\). Therefore, \(\cos 2\theta = 2\cos^2\theta - 1\). In our case, replace \(\cos^2 2\theta - \sin^2 2\theta\) using this identity:\[\cos^2 2\theta - \sin^2 2\theta = \cos 4\theta.\]

Step 3: Simplify the Right-Hand Side

The expression on the right-hand side is \(2 \cos^2 2\theta - 1\). We know from the identity that \(2\cos^2\theta - 1 = \cos 2\theta\). Here, replace \(\cos 4\theta\) using this identity:\[\cos 4\theta = 2 \cos^2 2\theta - 1.\]

Step 4: Verify the Identity

Now both sides become \(\cos 4\theta\), hence they are equal.\[\cos^2 2\theta - \sin^2 2\theta = 2 \cos^2 2\theta - 1.\]This confirms that the identity is verified by transforming one side into the other.

Key Concepts

Double Angle Formulas

Double angle formulas are essential tools in trigonometry. They allow the simplification and transformation of expressions involving trigonometric functions into more manageable forms.
The formula for the cosine of a double angle is particularly useful. It is given by

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• It can also be expressed as \( \cos 2\theta = 2\cos^2 \theta - 1 \) or \( \cos 2\theta = 1 - 2\sin^2 \theta \)

The beauty of these formulas is their versatility. They can transform trigonometric expressions into different, often simpler forms.
This makes solving trigonometric equations and proving identities much easier. For example, we used the identity \( \cos 2\theta \) to rewrite and verify the original identity in the problem as \( \cos^2 2\theta - \sin^2 2\theta = 2\cos^2 2\theta - 1 \).
Understanding these formulas is crucial for anyone working with trigonometric functions.

Cosine

Cosine is one of the primary trigonometric functions and is crucial in the study of geometry. Represented as \( \cos \), it measures the ratio of the adjacent side to the hypotenuse in a right triangle.
In the context of angles, cosine is extremely helpful. It describes how the function changes as the angle increases. The cosine function varies between -1 and 1, making it periodic with a predictably repeating pattern.

• At 0 degrees or radians, \( \cos \) is 1.
• At 90 degrees, \( \cos \) is 0.
• At 180 degrees, \( \cos \) is -1.
• At 270 degrees, \( \cos \) returns to 0.
• Finally, at 360 degrees, \( \cos \) returns to 1.

Cosine is crucial when applying double angle formulas. For instance, knowing \( \cos 2\theta \) as \( 2\cos^2 \theta - 1 \) simplifies problems substantially.
By altering the angle measure, previously complex expressions can become more approachable and easier to solve.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and require solutions for angles. Solving these equations can be daunting without understanding identities and forms.

Basic Steps

Start by identifying known identities, like double angle formulas. These formulas can transform and simplify equations.
Also, use algebraic techniques like factoring and expanding trigonometric expressions to isolate the variable angle.

Applications of Identities

Using identities brings another level of simplification, allowing you to see relationships between different equations. In our exercise, we verified an identity by recognizing and applying the double-angle formula.
This transformed the expression to a recognizable identity, making verification straightforward. By mastering these techniques and applying the right identities, many seemingly complex trigonometric equations become manageable. This process enhances the understanding and ability to handle a wide range of mathematical problems effectively.