Problem 27 - Algebra and Trigonometry

Question

Verify each identity. $$ \sin ^{2} x+\cos 2 x=\cos ^{2} x $$

Step-by-Step Solution

Answer

The identity \(\sin ^{2} x+\cos 2 x=\cos ^{2} x\) is verified by substituting \(\cos 2x\) with \(1 - 2\sin^{2} x\) and using the Pythagorean Trigonometric Identity.

Step 1: Substitute for \(\cos 2x\)

The identity \(\cos 2x = 1 - 2\sin^{2} x\) is used to replace \(\cos 2x\). The equation will then become \(\sin^{2} x + 1 - 2\sin^{2} x = \cos^{2} x\).

Step 2: Simplify the expression

Now, simplify the left-hand side of the equation by combining like terms. It will become \(1 - \sin^{2} x = \cos^{2} x\).

Step 3: Apply the Pythagorean Trigonometric Identity

Finally, note that \(\cos^{2} x\) is equivalent to \(1 - \sin^{2} x\) based on the Pythagorean Trigonometric Identity. Therefore, the equation \(1 - \sin^{2} x = \cos^{2} x\) is a valid identity, and the original equation is verified.

Key Concepts

Pythagorean Trigonometric Identity

Understanding trigonometric identities is crucial for solving a wide range of mathematical problems, particularly when studying trigonometry. The Pythagorean Trigonometric Identity is one of the foundational identities in this field and is represented by the equation: \[ \(\sin^2 x\) + \(\cos^2 x\) = 1 \.\] This identity is derived from the Pythagorean theorem related to the proportions of the sides of a right-angled triangle.

The essence of this identity is that the sum of the squares of the sine and cosine of an angle is always equal to 1. This is true for any angle \(x\), and it serves as a fundamental relationship between these two trigonometric functions. When solving trigonometry problems, converting one function to another using the Pythagorean identity can greatly simplify the process. It is particularly useful when verifying other identities, as we see in the given example.

Cosine Double Angle Identity

Expanding our toolkit of trigonometric identities, the cosine double angle identities are extremely beneficial for simplifying expressions involving trigonometric functions. There are a few variations, but one commonly used form is: \[ \cos 2x = 1 - 2\sin^2 x \.\] Other forms include: \[ \cos 2x = 2\cos^2 x - 1 \.\] and \[ \cos 2x = \cos^2 x - \sin^2 x \.\]

The choice of which form to use is context-dependent and can depend on the functions present in the equation you're working with. In the context of our example problem, the first variant is used to substitute in and thus is integral to the verification process of the original trigonometric identity. Understanding these identities helps to transform and simplify trigonometric expressions, a key skill in solving more complex problems.

Trigonometry Problem Solving

Tackling a trigonometry problem successfully often involves recognizing which identities are applicable and understanding how to manipulate and apply them effectively. Effective problem-solving in trigonometry is not just about memorizing identities but also understanding their interrelationships and practical applications.

When approaching a problem, it's helpful to:

• Identify known values and the relationships between angles and sides
• Choose the most appropriate trigonometric identities based on the given information
• Simplify expressions by substituting equivalent identities
• Work through algebraic manipulation patiently and carefully to rearrange and simplify the equation

In the example provided, the solution takes advantage of the cosine double angle identity to rewrite the expression, then simplifies it using algebra, and finally applies the Pythagorean Trigonometric Identity to verify the original statement. Such a methodical approach is not just useful for verifying identities but is also critical when solving a wide range of trigonometric equations and modeling real-world scenarios.